4.8E: Mazoezi ya Sehemu ya 4.8

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Katika mazoezi 1 - 6, tathmini kikomo.

1) Tathmini kikomo\(\displaystyle \lim_{x→∞}\frac{e^x}{x}\).

2) Tathmini kikomo\(\displaystyle \lim_{x→∞}\frac{e^x}{x^k}\).

Jibu: \(\displaystyle \lim_{x→∞}\frac{e^x}{x^k} \quad = \quad ∞\)

3) Tathmini kikomo\(\displaystyle \lim_{x→∞}\frac{\ln x}{x^k}\).

4) Tathmini kikomo\(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2}\).

Jibu: \(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2} \quad = \quad \frac{1}{2a}\)

5. Tathmini kikomo\(\displaystyle \lim_{x→a}\frac{x−a}{x^3−a^3}\).

6. Tathmini kikomo\(\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n}\).

Jibu: \(\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n} \quad = \quad \frac{1}{na^{n−1}}\)

Katika mazoezi 7 - 11, onyesha kama unaweza kutumia utawala wa L'Hôpital moja kwa moja. Eleza kwa nini au kwa nini. Kisha, onyesha kama kuna njia fulani unaweza kubadilisha kikomo ili uweze kutumia utawala wa L'Hôpital.

7)\(\displaystyle \lim_{x→0^+}x^2\ln x\)

8)\(\displaystyle \lim_{x→∞}x^{1/x}\)

Jibu: Haiwezi kuomba moja kwa moja; tumia logarithms

9)\(\displaystyle \lim_{x→0}x^{2/x}\)

10)\(\displaystyle \lim_{x→0}\frac{x^2}{1/x}\)

Jibu: Haiwezi kuomba moja kwa moja; kuandika upya kama\(\displaystyle \lim_{x→0}x^3\)

11)\(\displaystyle \lim_{x→∞}\frac{e^x}{x}\)

Katika mazoezi 12 - 40, tathmini mipaka na utawala wa L'Hôpital au mbinu zilizojifunza hapo awali.

12)\(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3}\)

Jibu: \(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3} \quad = \quad 6\)

13)\(\displaystyle \lim_{x→3}\frac{x^2−9}{x+3}\)

14)\(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x}\)

Jibu: \(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x} \quad = \quad -2\)

15)\(\displaystyle \lim_{x→π/2}\frac{\cos x}{\frac{π}{2}−x}\)

16)\(\displaystyle \lim_{x→π}\frac{x−π}{\sin x}\)

Jibu: \(\displaystyle \lim_{x→π}\frac{x−π}{\sin x} \quad = \quad -1\)

17)\(\displaystyle \lim_{x→1}\frac{x−1}{\sin x}\)

18)\(\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x}\)

Jibu: \(\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x} \quad = \quad n\)

19)\(\displaystyle \lim_{x→0}\frac{(1+x)^n−1−nx}{x^2}\)

20)\(\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3}\)

Jibu: \(\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3} \quad = \quad −\frac{1}{2}\)

21)\(\displaystyle \lim_{x→0}\frac{\sqrt{1+x}−\sqrt{1−x}}{x}\)

22)\(\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2}\)

Jibu: \(\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2} \quad = \quad \frac{1}{2}\)

23)\(\displaystyle \lim_{x→0}\frac{\tan x}{\sqrt{x}}\)

24)\(\displaystyle \lim_{x→1}\frac{x-1}{\ln x}\)

Jibu: \(\displaystyle \lim_{x→1}\frac{x-1}{\ln x} \quad = \quad 1\)

25)\(\displaystyle \lim_{x→0}\,(x+1)^{1/x}\)

26)\(\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1}\)

Jibu: \(\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1} \quad = \quad \frac{1}{6}\)

27)\(\displaystyle \lim_{x→0^+}x^{2x}\)

28)\(\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right)\)

Jibu: \(\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 1\)

29)\(\displaystyle \lim_{x→0}\frac{\sin x−x}{x^2}\)

30)\(\displaystyle \lim_{x→0^+}x\ln\left(x^4\right)\)

Jibu: \(\displaystyle \lim_{x→0^+}x\ln\left(x^4\right) \quad = \quad 0\)

31)\(\displaystyle \lim_{x→∞}(x−e^x)\)

32)\(\displaystyle \lim_{x→∞}x^2e^{−x}\)

Jibu: \(\displaystyle \lim_{x→∞}x^2e^{−x} \quad = \quad 0\)

33)\(\displaystyle \lim_{x→0}\frac{3^x−2^x}{x}\)

34)\(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x}\)

Jibu: \(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x} \quad = \quad -1\)

35)\(\displaystyle \lim_{x→π/4}(1−\tan x)\cot x\)

36)\(\displaystyle \lim_{x→∞}xe^{1/x}\)

Jibu: \(\displaystyle \lim_{x→∞}xe^{1/x} \quad = \quad ∞\)

37)\(\displaystyle \lim_{x→0}x^{1/\cos x}\)

38)\(\displaystyle \lim_{x→0^{+} }x^{1/x}\)

Jibu: \(\displaystyle \lim_{x→0^{+} }x^{1/x} \quad = \quad 0\)

39)\(\displaystyle \lim_{x→0}\left(1−\frac{1}{x}\right)^x\)

40)\(\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x\)

Jibu: \(\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x \quad = \quad \frac{1}{e}\)

Kwa mazoezi 41 - 50, tumia calculator ili kuchapisha kazi na ukadiria thamani ya kikomo, halafu utumie utawala wa L'Hôpital ili upate kikomo moja kwa moja.

41) [T]\(\displaystyle \lim_{x→0}\frac{e^x−1}{x}\)

42) [T]\(\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right)\)

Jibu: \(\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 0\)

43) [T]\(\displaystyle \lim_{x→1}\frac{x−1}{1−\cos(πx)}\)

44) [T]\(\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1}\)

Jibu: \(\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1} \quad = \quad 1\)

45) [T]\(\displaystyle \lim_{x→1}\frac{(x−1)^2}{\ln x}\)

46) [T]\(\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x}\)

Jibu: \(\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x} \quad = \quad 0\)

47) [T]\(\displaystyle \lim_{x→0}\left(\csc x−\frac{1}{x}\right)\)

48) [T]\(\displaystyle \lim_{x→0^+}\tan\left(x^x\right)\)

Jibu: \(\displaystyle \lim_{x→0^+}\tan\left(x^x\right) \quad = \quad \tan 1\)

49) [T]\(\displaystyle \lim_{x→0^+}\frac{\ln x}{\sin x}\)

50) [T]\(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x}\)

Jibu: \(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x} \quad = \quad 2\)