Limits and Continuity: part 8: IITJEE Math: Tutorial Problems for Practice

Problem 1:

Evaluate: \lim_{x \rightarrow \frac{\pi}{6}} \frac{2-\sqrt{3}\cos{x}-\sin{x}}{(6x-\pi)^{2}}

Problem 2:

Evaluate: \lim_{x \rightarrow 1} \frac{1-x^{2}}{\sin{(x\pi)}}

Problem 3:

Evaluate: \lim_{x \rightarrow 1}\frac{\cot{(\frac{\pi}{2}x)}}{x-1}

Problem 4:

Evaluate: \lim_{x \rightarrow 0} (1+\frac{4x}{5})^{\frac{10}{x}}

Problem 5:

Evaluate: \lim_{x \rightarrow 0} (\frac{1+ax}{1+bx})^{\frac{1}{x}}

Problem 6:

Evaluate: \lim_{x \rightarrow 0} (\frac{5+x}{5-x})^{\frac{1}{x}}

Problem 7:

Evaluate: \lim_{x \rightarrow 0} (\frac{4-3x}{4+5x})^{\frac{1}{x}}

Problem 8:

Evaluate: \lim_{x \rightarrow \infty} (1+ \frac{4}{n})^{3n}

Problem 9:

Evaluate: \lim_{x \rightarrow 1} \frac{\log{(2-x)}}{\sqrt{(3+x)}-2}

Problem 10:

Evaluate: \lim_{x \rightarrow 0} \frac{a^{x}-b^{x}}{3\sin{x} - \sin{(5x)}}

Problem 11:

Evaluate: \lim_{x \rightarrow 0} \frac{x \tan{x}}{e^{x}+e^{-x}-2}

Problem 12:

Evaluate: \lim_{x \rightarrow \frac{\pi}{4}} \frac{e^{(x - \frac{\pi}{4})}-1}{\cos{x} - \sin{x}}

Problem 13:

Evaluate: \lim_{x \rightarrow \frac{\pi}{2}} \frac{3^{(x - \frac{\pi}{2})} - 6^{(x - \frac{\pi}{2})}}{\cos{x}}

Problem 14:

Evaluate: \lim_{x \rightarrow 0} \frac{a^{3x}-a^{2x}-a^{x}+1}{x^{2}}

Problem 15:

Evaluate: \lim_{x \rightarrow 0} \frac{a^{x}+b^{x}-2^{(x+1)}}{x}

Problem 16:

Evaluate: \lim_{x \rightarrow \frac{\pi}{2}} \frac{2^{-\cos{x}}-1}{x(x - \frac{\pi}{2})}

Problem 17:

Evaluate: \lim_{\theta \rightarrow 0} \frac{3-4\cos{\theta}+\cos{2\theta}}{\theta^{4}}

Problem 18:

Evaluate: \lim_{x \rightarrow a} \frac{x \sin{a} - a \sin{x}}{x-a}

Problem 19:

Evaluate: \lim_{x \rightarrow 0} \frac{(27)^{x}-9^{x}-3^{x}+1}{\sqrt{2} - \sqrt{(1+\cos{x})}}

Problem 20:

Evaluate: \lim_{x \rightarrow 0} \frac{(5^{x}-2^{x})x}{\cos{5x} - \cos{3x}}

Problem 21:

Evaluate: \lim_{x \rightarrow 0} \frac{(3^{x}-1)^{2}}{2(1-\cos{x}) \log{(2+x)}}

Problem 22:

Evaluate: \lim_{x \rightarrow 1} \frac{\cos{(x \pi)} + \sin {(\frac{\pi}{2})x}}{(x-1)^{2}}

Problem 23:

Evaluate: \lim_{\theta \rightarrow \frac{\pi}{2}} \frac{\sin {\theta} + \cos{2 \theta}}{(\pi - 2 \theta)^{2}}

Problem 24:

Evaluate: \lim_{x \rightarrow \frac{1}{2}} \frac{2x^{2}+x-1}{4x^{2}-1+\sin{(2x-1)}}

Problem 25:

Evaluate: \lim_{x \rightarrow \infty} (\frac{2x+1}{2x-1})^{x+4}

Problem 26:

Evaluate: \lim_{x \rightarrow 0} \frac{e^{x} -2\cos{x} + e^{-x}}{x \sin{x}}

Problem 27:

Evaluate: \lim_{x \rightarrow 0} \frac{x^{2}}{\tan{x}} \sin{(\frac{1}{x})}

Problem 28:

Evaluate: \lim_{x \rightarrow 4} \frac{(\cos{\alpha})^{x} - (\sin{\alpha})^{x} -\cos{2\alpha}}{x-4}

There is one of the four possible answers:

(i) \log {(\frac{(\cos{\alpha})^{\cos^{-4}(\alpha)}}{(\sin{\alpha})^{\sin^{4}{(\alpha)}}})}

(ii) \log{(\frac{(\cos{\alpha})^{\cos^{4}{(\alpha)}}}{ (\sin{(\alpha)})^{\sin^{4}{(\alpha)}}})}

(iii) \log{(\frac{(\sin{\alpha})^{\sin^{4}{(\alpha)}}}{(\cos{(\alpha)})^{\cos^{4}{(\alpha)}}})}

(iv) \log{(\frac{(\sin{\alpha})^{\sin^{4}{\alpha}}}{(\cos{\alpha})^{\cos^{-4}{\alpha}}})}

Problem 29:

The values of A and B for f(x) to be continuous at x=0 where

f(x) = \frac{10^{x}+7^{x}-14^{x}-5^{x}}{1-\cos{x}} when x \neq 0

f(x) = \log{A} . \log{B} when x=0 are

(i) \frac{20}{7}, 1 (ii) \frac{10}{7}, 2 (iii) \frac{13}{7}, 1 (iv) \frac{5}{7}, 4

Problem 30:

If f(x) = \frac{\sqrt{1+\cos{x}}-1}{(\pi - x)^{2}} when x \neq \pi

and f(x) = k when x=\pi

Find the value of k for which f(x) is continuous at \pi.

Regards,

Nalin Pithwa

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