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2.2E: Mazoezi ya Sehemu ya 2.2

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    178956

    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    Ufafanuzi wa Intuitive wa

    Kwa mazoezi 1 - 2, fikiria kazi\(f(x)=\dfrac{x^2−1}{|x−1|}\).

    1) [T] Kukamilisha meza ifuatayo kwa ajili ya kazi. Pindua ufumbuzi wako kwa maeneo manne ya decimal.

    \(x\) \(f(x)\) \(x\) \(f(x)\)
    \ (x\)” style="text-align:katikati; "> 0.9 \ (f (x)\)” style="Nakala-align:center; ">a. \ (x\)” style="text-align:katikati; "> 1.1 \ (f (x)\)” style="Nakala-align:center; ">e.
    \ (x\)” style="text-align:katikati; "> 0.99 \ (f (x)\)” style="Nakala-align:center; "> b. \ (x\)” style="Nakala-align:katikati; "> 1.01 \ (f (x)\)” style="Nakala-align:center; ">f.
    \ (x\)” style="Nakala-align:katikati; "> 0.999 \ (f (x)\)” style="Nakala-align:center; ">c. \ (x\)” style="text-align:katikati; "> 1.001 \ (f (x)\)” style="Nakala-align:center; ">g.
    \ (x\)” style="Nakala-align:katikati; "> 0.9999 \ (f (x)\)” style="Nakala-align:center; ">d. \ (x\)” style="Nakala-align:katikati; "> 1.0001 \ (f (x)\)” style="Nakala-align:center; ">h.

    2) Matokeo yako katika zoezi la awali yanaonyesha nini kuhusu kikomo cha upande mmoja\(\displaystyle \lim_{x→1}f(x)\)? Eleza majibu yako.

    Jibu

    \(\displaystyle \lim_{x \to 1}f(x)\)haipo kwa sababu\(\displaystyle \lim_{x \to 1^−}f(x)=−2≠\lim_{x \to 1^+}f(x)=2\).

    Kwa mazoezi 3 - 5, fikiria kazi\(f(x)=(1+x)^{1/x}\).

    3) [T] Fanya meza kuonyesha maadili ya\(f\) kwa\(x=−0.01,\;−0.001,\;−0.0001,\;−0.00001\) na kwa\(x=0.01,\;0.001,\;0.0001,\;0.00001\). Pindua ufumbuzi wako kwa maeneo tano ya decimal.

    \(x\) \(f(x)\) \(x\) \(f(x)\)
    \ (x\)” style="Nakala-align:katikati; "> -0.01 \ (f (x)\)” style="Nakala-align:center; ">a. \ (x\)” style="text-align:katikati; "> 0.01 \ (f (x)\)” style="Nakala-align:center; ">e.
    \ (x\)” style="text-align:katikati; "> -0.001 \ (f (x)\)” style="Nakala-align:center; "> b. \ (x\)” style="text-align:katikati; "> 0.001 \ (f (x)\)” style="Nakala-align:center; ">f.
    \ (x\)” style="Nakala-align:katikati; "> -0.0001 \ (f (x)\)” style="Nakala-align:center; ">c. \ (x\)” style="Nakala-align:katikati; "> 0.0001 \ (f (x)\)” style="Nakala-align:center; ">g.
    \ (x\)” style="text-align:katikati; "> -0.00001 \ (f (x)\)” style="Nakala-align:center; ">d. \ (x\)” style="text-align:katikati; "> 0.00001 \ (f (x)\)” style="Nakala-align:center; ">h.

    4) Jedwali la maadili katika zoezi la awali linaonyesha nini kuhusu kazi\(f(x)=(1+x)^{1/x}\)?

    Jibu
    \(\displaystyle \lim_{x \to 0}(1+x)^{1/x}\approx 2.7183\).

    5) Ni mara kwa mara ya hisabati ambayo maadili katika zoezi la awali yanaonekana kuwa inakaribia? Hii ni kikomo halisi hapa.

    Katika mazoezi ya 6 - 8, tumia maadili yaliyotolewa ili kuanzisha meza ili kutathmini mipaka. Pindua ufumbuzi wako kwa maeneo nane ya decimal.

    6) [T]\(\displaystyle \lim_{x \to 0}\frac{\sin 2x}{x};\quad ±0.1,\; ±0.01, \; ±0.001, \;±.0001\)

    \(x\) \(\frac{\sin 2x}{x}\) \(x\) \(\frac{\sin 2x}{x}\)
    \ (x\)” style="text-align:katikati; "> -0.1 \ (\ frac {\ sin 2x} {x}\)” style="text-align:katikati; ">a. \ (x\)” style="text-align:katikati; "> 0.1 \ (\ frac {\ sin 2x} {x}\)” style="text-align:katikati; ">e.
    \ (x\)” style="Nakala-align:katikati; "> -0.01 \ (\ frac {\ sin 2x} {x}\)” style="text-align:katikati; ">b. \ (x\)” style="text-align:katikati; "> 0.01 \ (\ frac {\ sin 2x} {x}\)” style="text-align:katikati; ">f.
    \ (x\)” style="text-align:katikati; "> -0.001 \ (\ frac {\ sin 2x} {x}\)” style="text-align:katikati; ">c. \ (x\)” style="text-align:katikati; "> 0.001 \ (\ frac {\ sin 2x} {x}\)” style="text-align:katikati; ">g.
    \ (x\)” style="Nakala-align:katikati; "> -0.0001 \ (\ frac {\ sin 2x} {x}\)” style="text-align:katikati; ">d. \ (x\)” style="Nakala-align:katikati; "> 0.0001 \ (\ frac {\ sin 2x} {x}\)” style="text-align:katikati; ">h.
    Jibu
    a. 1.98669331; b. 1.99986667; c. 1.99999867; d 1.99999999; e 1.98669331; f. 1.99986667; g 1.99999867; h 1.9999999999;
    \(\displaystyle \lim_{x \to 0}\frac{\sin 2x}{x}=2\)

    7) [T]\(\displaystyle \lim_{x \to 0}\frac{\sin 3x}{x} ±0.1, \; ±0.01, \; ±0.001, \; ±0.0001\)

    \(x\) \(\frac{\sin 3x}{x}\) \(x\) \(\frac{\sin 3x}{x}\)
    \ (x\)” style="text-align:katikati; "> -0.1 \ (\ frac {\ sin 3x} {x}\)” style="text-align:katikati; ">a. \ (x\)” style="text-align:katikati; "> 0.1 \ (\ frac {\ sin 3x} {x}\)” style="text-align:center; ">e.
    \ (x\)” style="Nakala-align:katikati; "> -0.01 \ (\ frac {\ sin 3x} {x}\)” style="text-align:katikati; ">b. \ (x\)” style="text-align:katikati; "> 0.01 \ (\ frac {\ sin 3x} {x}\)” style="text-align:katikati; ">f.
    \ (x\)” style="text-align:katikati; "> -0.001 \ (\ frac {\ sin 3x} {x}\)” style="text-align:katikati; ">c. \ (x\)” style="text-align:katikati; "> 0.001 \ (\ frac {\ sin 3x} {x}\)” style="text-align:katikati; ">g.
    \ (x\)” style="Nakala-align:katikati; "> -0.0001 \ (\ frac {\ sin 3x} {x}\)” style="text-align:katikati; ">d. \ (x\)” style="Nakala-align:katikati; "> 0.0001 \ (\ frac {\ sin 3x} {x}\)” style="text-align:katikati; ">h.

    8) Tumia mazoezi mawili yaliyotangulia kwa dhana (nadhani) thamani ya kikomo kinachofuata:\(\displaystyle \lim_{x \to 0}\frac{\sin ax}{x}\) kwa\(a\), thamani halisi halisi.

    Jibu
    \(\displaystyle \lim_{x \to 0}\frac{\sin ax}{x}=a\)

    [T] Katika mazoezi 9 - 14, weka meza ya maadili ili kupata kikomo kilichoonyeshwa. Pande zote kwa tarakimu nane muhimu.

    9)\(\displaystyle \lim_{x \to 2}\frac{x^2−4}{x^2+x−6}\)

    \(x\) \(\frac{x^2−4}{x^2+x−6}\) \(x\) \(\frac{x^2−4}{x^2+x−6}\)
    \ (x\)” style="Nakala-align:katikati; "> 1.9 \ (\ frac {x^2,1-4} {x^2+x-6}\)” style="Nakala-align:katikati; ">a. \ (x\)” style="text-align:katikati; "> 2.1 \ (\ frac {x^2,1-4} {x^2+x-6}\)” style="Nakala-align:katikati; ">e.
    \ (x\)” style="text-align:katikati; "> 1.99 \ (\ frac {x^2,1-4} {x^2+x-6}\)” style="Nakala-align:katikati; ">b. \ (x\)” style="Nakala-align:katikati; "> 2.01 \ (\ frac {x^2,1-4} {x^2+x-6}\)” style="Nakala-align:katikati; ">f.
    \ (x\)” style="Nakala-align:katikati; "> 1.999 \ (\ frac {x^2,1-4} {x^2+x-6}\)” style="Nakala-align:katikati; ">c. \ (x\)” style="text-align:katikati; "> 2.001 \ (\ frac {x^2,1-4} {x^2+x-6}\)” style="Nakala-align:katikati; ">g.
    \ (x\)” style="Nakala-align:katikati; "> 1.9999 \ (\ frac {x^2,1-4} {x^2+x-6}\)” style="Nakala-align:katikati; ">d. \ (x\)” style="Nakala-align:katikati; "> 2.0001 \ (\ frac {x^2,1-4} {x^2+x-6}\)” style="Nakala-align:katikati; ">h.

    10)\(\displaystyle \lim_{x \to 1}(1−2x)\)

    \(x\) \(1−2x\) \(x\) \(1−2x\)
    \ (x\)” style="text-align:katikati; "> 0.9 \ (1,12x\)” style="Nakala-align:katikati; ">a. \ (x\)” style="text-align:katikati; "> 1.1 \ (1,12x\)” style="Nakala-align:katikati; ">e.
    \ (x\)” style="text-align:katikati; "> 0.99 \ (1,12x\)” style="Nakala-align:katikati; "> b. \ (x\)” style="Nakala-align:katikati; "> 1.01 \ (1,12x\)” style="Nakala-align:katikati; ">f.
    \ (x\)” style="Nakala-align:katikati; "> 0.999 \ (1,12x\)” style="Nakala-align:katikati; ">c. \ (x\)” style="text-align:katikati; "> 1.001 \ (1,12x\)” style="Nakala-align:katikati; ">g.
    \ (x\)” style="Nakala-align:katikati; "> 0.9999 \ (1,12x\)” style="Nakala-align:katikati; ">d. \ (x\)” style="Nakala-align:katikati; "> 1.0001 \ (1,12x\)” style="Nakala-align:katikati; ">h.
    Jibu
    a. -0.80000000; b. -0.98000000; c. -0.99800000; d. -0.99980000; e. -1.2000000; f. -1.0200000; g. -1.0020000; h. -1.0002000;
    \( \displaystyle \lim_{x \to 1}(1−2x)=−1\)

    11)\(\displaystyle \lim_{x \to 0}\frac{5}{1−e^{1/x}}\)

    \(x\) \(\frac{5}{1−e^{1/x}}\) \(x\) \(\frac{5}{1−e^{1/x}}\)
    \ (x\)” style="text-align:katikati; "> -0.1 \ (\ frac {5} {1,1e-e^ {1/x}}\)” style="Nakala-align:katikati; ">a. \ (x\)” style="text-align:katikati; "> 0.1 \ (\ frac {5} {1,1e-e^ {1/x}}\)” style="Nakala-align:katikati; ">e.
    \ (x\)” style="Nakala-align:katikati; "> -0.01 \ (\ frac {5} {1,1e-e^ {1/x}}\)” style="Nakala-align:katikati; ">b. \ (x\)” style="text-align:katikati; "> 0.01 \ (\ frac {5} {1,1e-e^ {1/x}}\)” style="Nakala-align:katikati; ">f.
    \ (x\)” style="text-align:katikati; "> -0.001 \ (\ frac {5} {1,1e-e^ {1/x}}\)” style="Nakala-align:katikati; ">c. \ (x\)” style="text-align:katikati; "> 0.001 \ (\ frac {5} {1,1e-e^ {1/x}}\)” style="Nakala-align:katikati; ">g.
    \ (x\)” style="Nakala-align:katikati; "> -0.0001 \ (\ frac {5} {1,1e-e^ {1/x}}\)” style="Nakala-align:katikati; ">d. \ (x\)” style="Nakala-align:katikati; "> 0.0001 \ (\ frac {5} {1,1e-e^ {1/x}}\)” style="Nakala-align:katikati; ">h.

    12)\(\displaystyle \lim_{z \to 0}\frac{z−1}{z^2(z+3)}\)

    \(z\) \(\frac{z−1}{z^2(z+3)}\) \(z\) \(\frac{z−1}{z^2(z+3)}\)
    \ (z\)” style="text-align:katikati; ">-0.1 \ (\ frac {z-1} {z^2 (z+3)}\)” style="Nakala-align:katikati; ">a. \ (z\)” style="text-align:katikati; "> 0.1 \ (\ frac {z-1} {z^2 (z+3)}\)” style="text-align:katikati; ">e.
    \ (z\)” style="Nakala-align:katikati; "> -0.01 \ (\ frac {z-1} {z^2 (z+3)}\)” style="Nakala-align:katikati; ">b. \ (z\)” style="text-align:katikati; "> 0.01 \ (\ frac {z-1} {z^2 (z+3)}\)” style="text-align:katikati; ">f.
    \ (z\)” style="text-align:katikati; "> -0.001 \ (\ frac {z-1} {z^2 (z+3)}\)” style="Nakala-align:katikati; ">c. \ (z\)” style="text-align:katikati; "> 0.001 \ (\ frac {z-1} {z^2 (z+3)}\)” style="text-align:katikati; ">g.
    \ (z\)” style="text-align:katikati; "> -0.0001 \ (\ frac {z-1} {z^2 (z+3)}\)” style="Nakala-align:katikati; ">d. \ (z\)” style="text-align:katikati; "> 0.0001 \ (\ frac {z-1} {z^2 (z+3)}\)” style="Nakala-align:katikati; ">h.
    Jibu
    a. -37.931034; b. -3377.9264; c. -333,777.93; d. -33,337,778; e. -29.032258; f. -3289.0365; g. -332,889.04; h. -33,328,889
    \( \displaystyle \lim_{x \to 0}\frac{z−1}{z^2(z+3)}=−∞\)

    13)\(\displaystyle \lim_{t \to 0^+}\frac{\cos t}{t}\)

    \(t\) \(\frac{\cos t}{t}\)
    \ (t\)” style="text-align:katikati; "> 0.1 \ (\ frac {\ cos t} {t}\)” style="Nakala-align:katikati; ">a.
    \ (t\)” style="Nakala-align:katikati; "> 0.01 \ (\ frac {\ cos t} {t}\)” style="Nakala-align:katikati; ">b.
    \ (t\)” style="text-align:katikati; "> 0.001 \ (\ frac {\ cos t} {t}\)” style="Nakala-align:katikati; ">c.
    \ (t\)” style="Nakala-align:katikati; "> 0.0001 \ (\ frac {\ cos t} {t}\)” style="Nakala-align:katikati; ">d.

    14)\(\displaystyle \lim_{x \to 2}\frac{1−\frac{2}{x}}{x^2−4}\)

    \(x\) \(\frac{1−\frac{2}{x}}{x^2−4}\) \(x\) \(\frac{1−\frac{2}{x}}{x^2−4}\)
    \ (x\)” style="Nakala-align:katikati; "> 1.9 \ (\ frac {1-\ frac {2} {x}} {x^2,14}\)” style="Nakala-align:katikati; ">a. \ (x\)” style="text-align:katikati; "> 2.1 \ (\ frac {1-\ frac {2} {x}} {x^2,14}\)” style="Nakala-align:katikati; ">e.
    \ (x\)” style="text-align:katikati; "> 1.99 \ (\ frac {1-\ frac {2} {x}} {x^2,14}\)” style="Nakala-align:katikati; ">b. \ (x\)” style="Nakala-align:katikati; "> 2.01 \ (\ frac {1-\ frac {2} {x}} {x^2,14}\)” style="Nakala-align:katikati; ">f.
    \ (x\)” style="Nakala-align:katikati; "> 1.999 \ (\ frac {1-\ frac {2} {x}} {x^2,14}\)” style="Nakala-align:katikati; ">c. \ (x\)” style="text-align:katikati; "> 2.001 \ (\ frac {1-\ frac {2} {x}} {x^2,14}\)” style="Nakala-align:katikati; ">g.
    \ (x\)” style="Nakala-align:katikati; "> 1.9999 \ (\ frac {1-\ frac {2} {x}} {x^2,14}\)” style="Nakala-align:katikati; ">d. \ (x\)” style="Nakala-align:katikati; "> 2.0001 \ (\ frac {1-\ frac {2} {x}} {x^2,14}\)” style="Nakala-align:katikati; ">h.
    Jibu
    a. 0.13495277; b. 0.12594300; c. 0.12509381; d. 0.12500938; e. 0.11614402; f. 0.12406794; g 0.12490631; h 0.12499063;
    \( \displaystyle ∴\lim_{x \to 2}\frac{1−\frac{2}{x}}{x^2−4}=0.1250=\frac{1}{8}\)

    [T] Katika mazoezi 15 - 16, weka meza ya maadili na pande zote hadi tarakimu nane muhimu. Kulingana na meza ya maadili, fanya nadhani kuhusu kile kikomo ni. Kisha, tumia calculator kwa graph kazi na kuamua kikomo. Je, dhana hiyo ilikuwa sahihi? Ikiwa sio, kwa nini njia ya meza inashindwa?

    15)\(\displaystyle \lim_{θ \to 0}\sin\left(\frac{π}{θ}\right)\)

    \(θ\) \(\sin\left(\frac{π}{θ}\right)\) \(θ\) \(\sin\left(\frac{π}{θ}\right)\)
    \ (\)” style="text-align:katikati; ">-0.1 \ (\ dhambi\ kushoto (\ frac {π} {η}\ haki)\)” style="text-align:center; ">a. \ (\)” style="text-align:katikati; "> 0.1 \ (\ sin\ kushoto (\ frac {π} {η}\ haki)\)” style="text-align:center; ">e.
    \ (\)” style="text-align:katikati; "> -0.01 \ (\ dhambi\ kushoto (\ frac {π} {η}\ haki)\)” style="text-align:katikati; ">b. \ (\)” style="text-align:katikati; "> 0.01 \ (\ sin\ kushoto (\ frac {π} {η}\ haki)\)” style="text-align:katikati; ">f.
    \ (\)” style="text-align:katikati; "> -0.001 \ (\ sin\ kushoto (\ frac {π} {η}\ haki)\)” style="text-align:katikati; ">c. \ (\)” style="text-align:katikati; "> 0.001 \ (\ sin\ kushoto (\ frac {π} {η}\ haki)\)” style="text-align:katikati; ">g.
    \ (\)” style="text-align:katikati; "> -0.0001 \ (\ sin\ kushoto (\ frac {π} {η}\ haki)\)” style="text-align:katikati; ">d. \ (\)” style="text-align:katikati; "> 0.0001 \ (\ sin\ kushoto (\ frac {π} {η}\ haki)\)” style="text-align:center; ">h.

    16)\(\displaystyle \lim_{α \to 0^+} \frac{1}{α}\cos\left(\frac{π}{α}\right)\)

    \(a\) \(\frac{1}{α}\cos\left(\frac{π}{α}\right)\)
    \ (a\)” style="text-align:katikati; "> 0.1 \ (\ frac {1} {α}\ cos\ kushoto (\ frac {π} {α}\ haki)\)” style="text-align:center; ">a.
    \ (a\)” style="Nakala-align:katikati; "> 0.01 \ (\ frac {1} {α}\ cos\ kushoto (\ frac {π} {α}\ haki)\)” style="text-align:center; ">b.
    \ (a\)” style="text-align:katikati; "> 0.001 \ (\ frac {1} {α}\ cos\ kushoto (\ frac {π} {α}\ haki)\)” style="text-align:center; ">c.
    \ (a\)” style="Nakala-align:katikati; "> 0.0001 \ (\ frac {1} {α}\ cos\ kushoto (\ frac {π} {α}\ haki)\)” style="text-align:center; ">d.
    Jibu

    a. 10.00000; b. 100.00000; c. 1000.0000; d. 10,000.000;
    Nadhani:\(\displaystyle \lim_{α→0^+}\frac{1}{α}\cos\left(\frac{π}{α}\right)=∞\);
    Kweli: DNE, tangu grafu inaonyesha kazi oscillates wildly kati ya maadili inakaribia infinity chanya na maadili inakaribia infinity hasi, kama thamani ya\(α\) mbinu \(0\)kutoka upande chanya.

    Grafu ya kazi (1/alpha) * cos (pi/alpha), ambayo oscillates upole mpaka muda [-.2, .2], ambapo oscillates haraka, kwenda infinity na infinity hasi kama inakaribia mhimili y.

    Katika mazoezi 17 - 20, fikiria grafu ya kazi\(y=f(x)\) iliyoonyeshwa hapa. Ni ipi kati ya kauli kuhusu\(y=f(x)\) ni kweli na ambayo ni ya uongo? Eleza kwa nini taarifa ni ya uongo.

    Grafu ya kazi ya kipande na makundi matatu na uhakika. Sehemu ya kwanza ni safu inayofungua juu na vertex saa (-8, -6). Kipeo hiki ni mduara wazi, na kuna mduara uliofungwa badala yake (-8, -3). Sehemu hiyo inaisha saa (-2,3), ambapo kuna mduara uliofungwa. Sehemu ya pili inaenea kwa urahisi hadi infinity pamoja na x=-2, inabadilisha mwelekeo kuongezeka kwa karibu (0,1.25), huongezeka hadi karibu (2.25, 3), na hupungua hadi (6,2), ambapo kuna mduara wazi. Sehemu ya mwisho huanza saa (6,5), huongezeka kidogo, na kisha hupungua katika quadrant nne, kuvuka mhimili x saa (10,0). Mabadiliko yote katika mwelekeo ni curves laini.

    17)\(\displaystyle \lim_{x→10}f(x)=0\)

    18)\(\displaystyle \lim_{x→−2^+}f(x)=3\)

    Jibu
    Uongo;\(\displaystyle \lim_{x→−2^+}f(x)=+∞\)

    19)\(\displaystyle \lim_{x→−8}f(x)=f(−8)\)

    20)\(\displaystyle \lim_{x→6}f(x)=5\)

    Jibu
    uongo;\(\displaystyle \lim_{x→6}f(x)\) DNE tangu\(\displaystyle \lim_{x→6^−}f(x)=2\) na\(\displaystyle \lim_{x→6^+}f(x)=5\).

    Katika mazoezi 21 - 25, tumia grafu inayofuata ya kazi\(y=f(x)\) ili kupata maadili, ikiwa inawezekana. Tathmini wakati inahitajika.

    Grafu ya kazi ya kipande na makundi mawili. Sehemu ya kwanza ipo kwa x <=1, na sehemu ya pili ipo kwa x 1. Sehemu ya kwanza ni sawa na mteremko wa 1 na huenda kupitia asili. Mwisho wake ni mduara uliofungwa kwenye (1,1). Sehemu ya pili pia ni sawa na mteremko wa -1. Inaanza na mduara wazi katika (1,2)." style="width: 417px; height: 424px;" width="417px" height="424px" src="https://math.libretexts.org/@api/dek...02_02_202.jpeg">

    21)\(\displaystyle \lim_{x→1^−}f(x)\)

    22)\(\displaystyle \lim_{x→1^+}f(x)\)

    Jibu
    \(2\)

    23)\(\displaystyle \lim_{x→1}f(x)\)

    24)\(\displaystyle \lim_{x→2}f(x)\)

    Jibu
    \(1\)

    25)\(f(1)\)

    Katika mazoezi 26 - 29, tumia grafu ya kazi\(y=f(x)\) iliyoonyeshwa hapa ili kupata maadili, ikiwa inawezekana. Tathmini wakati inahitajika.

    Grafu ya kazi ya kipande na makundi mawili. Ya kwanza ni kazi ya mstari kwa x <0. Kuna mduara wazi (0,1), na mteremko wake ni -1. Sehemu ya pili ni nusu sahihi ya parabola kufungua juu. Kipeo chake ni mduara uliofungwa kwenye (0, -4), na unapitia hatua (2,0).

    26)\(\displaystyle \lim_{x→0^−}f(x)\)

    Jibu
    \(1\)

    27)\(\displaystyle \lim_{x→0^+}f(x)\)

    28)\(\displaystyle \lim_{x→0}f(x)\)

    Jibu
    DNE

    29)\(\displaystyle \lim_{x→2}f(x)\)

    Katika mazoezi 30 - 35, tumia grafu ya kazi\(y=f(x)\) iliyoonyeshwa hapa ili kupata maadili, ikiwa inawezekana. Tathmini wakati inahitajika.

    Grafu ya kazi ya kipande na makundi matatu, yote ya mstari. Ya kwanza ipo kwa x <-2, ina mteremko wa 1, na inaisha kwenye mduara wazi saa (-2, 0). Ya pili ipo juu ya muda [-2, 2], ina mteremko wa -1, hupitia asili, na imefungwa miduara kwenye mwisho wake (-2, 2) na (2, -2). Ya tatu ipo kwa x2, ina mteremko wa 1, na huanza kwenye mduara wazi (2,2)." style="width: 417px; height: 424px;" width="417px" height="424px" src="https://math.libretexts.org/@api/dek...02_02_204.jpeg">

    30)\(\displaystyle \lim_{x→−2^−}f(x)\)

    Jibu
    \(0\)

    31)\(\displaystyle \lim_{x→−2^+}f(x)\)

    32)\(\displaystyle \lim_{x→−2}f(x)\)

    Jibu
    DNE

    33)\(\displaystyle \lim_{x→2^−}f(x)\)

    34)\(\displaystyle \lim_{x→2^+}f(x)\)

    Jibu
    \(2\)

    35)\(\displaystyle \lim_{x→2}f(x)\)

    Katika mazoezi 36 - 38, tumia grafu ya kazi\(y=g(x)\) iliyoonyeshwa hapa ili kupata maadili, ikiwa inawezekana. Tathmini wakati inahitajika.

    Grafu ya kazi ya kipande na makundi mawili. Ya kwanza ipo kwa x=0 na ni nusu ya kushoto ya parabola ya ufunguzi wa juu na kipeo kwenye mduara uliofungwa (0,3). Ya pili ipo kwa x>0 na ni nusu sahihi ya parabola ya ufunguzi wa chini na vertex kwenye mduara wazi (0,0)." style="width: 417px; height: 424px;" width="417px" height="424px" src="https://math.libretexts.org/@api/dek...02_02_205.jpeg">

    36)\(\displaystyle \lim_{x→0^−}g(x)\)

    Jibu
    \(3\)

    37)\(\displaystyle \lim_{x→0^+}g(x)\)

    38)\(\displaystyle \lim_{x→0}g(x)\)

    Jibu
    DNE

    Katika mazoezi 39 - 41, tumia grafu ya kazi\(y=h(x)\) iliyoonyeshwa hapa ili kupata maadili, ikiwa inawezekana. Tathmini wakati inahitajika.

    Grafu ya kazi na curves mbili inakaribia 0 kutoka quadrant 1 na roboduara 3. Curve katika roboduara moja inaonekana kuwa nusu ya juu ya ufunguzi wa parabola upande wa kulia wa mhimili y pamoja na mhimili x na vertex katika asili. Curve katika roboduara tatu inaonekana kuwa nusu ya kushoto ya parabola inayofunguliwa chini na kipeo katika asili.

    39)\(\displaystyle \lim_{x→0^−}h(x)\)

    40)\(\displaystyle \lim_{x→0^+}h(x)\)

    Jibu
    \(0\)

    41)\(\displaystyle \lim_{x→0}h(x)\)

    Katika mazoezi 42 - 46, tumia grafu ya kazi\(y=f(x)\) iliyoonyeshwa hapa ili kupata maadili, ikiwa inawezekana. Tathmini wakati inahitajika.

    Grafu yenye safu na uhakika. Hatua ni mduara uliofungwa (0, -2). Curve ni sehemu ya juu ya ufunguzi parabola na kipeo katika (1, -1). Ipo kwa x 0, na kuna mduara uliofungwa katika asili." style="width: 417px; height: 424px;" width="417px" height="424px" src="https://math.libretexts.org/@api/dek...02_02_207.jpeg">

    42)\(\displaystyle \lim_{x→0^−}f(x)\)

    Jibu
    \(-2\)

    43)\(\displaystyle \lim_{x→0^+}f(x)\)

    44)\(\displaystyle \lim_{x→0}f(x)\)

    Jibu
    DNE

    45)\(\displaystyle \lim_{x→1}f(x)\)

    46)\(\displaystyle \lim_{x→2}f(x)\)

    Jibu
    \(0\)

    mipaka usio

    Katika mazoezi 47 - 51, mchoro grafu ya kazi na mali zilizopewa.

    47)\(\displaystyle \lim_{x→2}f(x)=1, \quad \lim_{x→4^−}f(x)=3, \quad \lim_{x→4^+}f(x)=6, \quad x=4\) haijafafanuliwa.

    48)\(\displaystyle \lim_{x→−∞}f(x)=0, \quad \lim_{x→−1^−}f(x)=−∞, \quad \lim_{x→−1^+}f(x)=∞,\quad \lim_{x→0}f(x)=f(0), \quad f(0)=1, \quad \lim_{x→∞}f(x)=−∞\)

    Jibu

    Majibu inaweza kutofautiana

    Grafu ya kazi ya kipande na makundi mawili. Sehemu ya kwanza iko katika quadrant tatu na haipatikani huenda kwenye infinity hasi kando ya mhimili y na 0 pamoja na mhimili x. Sehemu ya pili ina curves mbili. Ya kwanza inaonekana kuwa nusu ya kushoto ya parabola ya ufunguzi wa juu na vertex saa (0,1). Ya pili inaonekana kuwa nusu sahihi ya parabola ya ufunguzi wa chini na vertex saa (0,1) pia.

    49)\(\displaystyle \lim_{x→−∞}f(x)=2, \quad \lim_{x→3^−}f(x)=−∞, \quad \lim_{x→3^+}f(x)=∞, \quad \lim_{x→∞}f(x)=2, \quad f(0)=-\frac{1}{3}\)

    50)\(\displaystyle \lim_{x→−∞}f(x)=2,\quad \lim_{x→−2}f(x)=−∞,\quad \lim_{x→∞}f(x)=2,\quad f(0)=0\)

    Jibu

    Jibu inaweza kutofautiana

    Grafu iliyo na curves mbili. Ya kwanza inakwenda 2 kwa njia isiyo ya kawaida pamoja na y = 2 na kwa infinity hasi pamoja x = -2. Ya pili inakwenda infinity hasi pamoja x=-2 na 2 pamoja y=2.

    51)\(\displaystyle \lim_{x→−∞}f(x)=0,\quad \lim_{x→−1^−}f(x)=∞,\quad \lim_{x→−1^+}f(x)=−∞, \quad f(0)=−1, \quad \lim_{x→1^−}f(x)=−∞, \quad \lim_{x→1^+}f(x)=∞, \quad \lim_{x→∞}f(x)=0\)

    52) Mawimbi ya mshtuko hutokea katika maombi mengi ya kimwili, kuanzia supernovas hadi mawimbi ya kupasuka. Grafu ya wiani wa wimbi la mshtuko kwa heshima na umbali,\(x\), inavyoonyeshwa hapa. Sisi ni hasa nia ya eneo la mbele ya mshtuko, iliyoandikwa\(X_{SF}\) kwenye mchoro.

    Grafu katika quadrant moja ya wiani wa mshtuko na pointi tatu zilizoandikwa: p1 na p2 kwenye mhimili y, na p1 p2, na xsf kwenye mhimili x. Inajumuisha y= p1 kutoka 0 hadi xsf, x = xsf kutoka y= p1 hadi y = p2, na y = p2 kwa maadili makubwa kuliko au sawa na xsf." style="width: 300px; height: 304px;" width="300px" height="304px" src="https://math.libretexts.org/@api/dek...avedensity.png">

    a. kutathmini\(\displaystyle \lim_{x→X_{SF}^+}ρ(x)\).

    b Tathmini\(\displaystyle \lim_{x→X_{SF}^−}ρ(x)\).

    c. kutathmini\(\displaystyle \lim_{x→X_{SF}}ρ(x)\). Eleza maana ya kimwili nyuma ya majibu yako.

    Jibu
    a.\(ρ_2\) b.\(ρ_1\) c. DNE isipokuwa\(ρ_1=ρ_2\). \(X_{SF}\)Unapokaribia kutoka kulia, uko katika eneo la juu-wiani wa mshtuko. Unapokaribia kutoka upande wa kushoto, hujapata “mshtuko” bado na una wiani wa chini.

    53) Kocha wa kufuatilia anatumia kamera yenye shutter ya haraka ili kukadiria nafasi ya mkimbiaji kwa heshima na wakati. Jedwali la maadili ya msimamo wa mwanariadha dhidi ya wakati hutolewa hapa, wapi\(x\) nafasi katika mita za mkimbiaji na\(t\) ni wakati kwa sekunde. Ni nini\(\displaystyle \lim_{t→2}x(t)\)? Ina maana gani kimwili?

    \(t(sec)\) \(x(m)\)
    \ (t (sec)\)” style="Nakala-align:center; "> 1.75 \ (x (m)\)” style="Nakala-align:center; "> 4.5
    \ (t (sec)\)” style="Nakala-align:center; "> 1.95 \ (x (m)\)” style="Nakala-align:center; "> 6.1
    \ (t (sec)\)” style="Nakala-align:center; "> 1.99 \ (x (m)\)” style="Nakala-align:katikati; "> 6.42
    \ (t (sec)\)” style="Nakala-align:center; "> 2.01 \ (x (m)\)” style="Nakala-align:center; "> 6.58
    \ (t (sec)\)” style="Nakala-align:center; "> 2.05 \ (x (m)\)” style="Nakala-align:center; "> 6.9
    \ (t (sec)\)” style="Nakala-align:center; "> 2.25 \ (x (m)\)” style="Nakala-align:center; "> 8.5