Limits and Continuity: part 7: IITJEE math: Tutorial Problems for Practice

Problem 1:

Find the value of the following limit: $\lim_{\theta \rightarrow \frac{\pi}{4}} \frac{2- cosec(\theta)*cosec(\theta)}{1-\cos{\theta}}$

Problem 2:

Find the value of the following limit: $\lim_{x \rightarrow 2}\frac{2x^{2}-7x+6}{5x^{2}-11x+2}$

Problem 3:

Find the value of the following limit: $\lim_{x \rightarrow 4} \frac{x^{4}-64x}{\sqrt{(x^{2}+9)}-5}$

Problem 4:

Find the value of the following limit: $\lim_{x \rightarrow 2} (\frac{1}{x-2} + \frac{6x}{8-x^{3}})$

Problem 5:

Find the value of the following limit: $\lim_{x \rightarrow \infty} \frac{4x^{4}-3x^{3}+2x^{2}-x+1}{3x^{4}-2x^{3}+x^{2}-x-7}$

Problem 6:

Find the value of the following limit: $\lim_{x \rightarrow \infty}(\sqrt{x^{2}+4x+5} -\sqrt{x^{2}+1})$

Problem 7:

Find the following limit: $\lim_{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$ where $f(x) = \sqrt{7-2x}$

Problem 8:

Evaluate: $\lim_{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots+x^{2n} -2n}{x-1}$ , where $n \in N$

Problem 9:

Evaluate: $\lim_{x \rightarrow 0} \frac{1-\cos{(2x)}}{\cos{(2x)}-\cos{(8x)}}$

Problem 10:

Evaluate: $\lim_{\theta \rightarrow 0} \frac{5\theta\cos{\theta}-2\sin{\theta}}{3\theta+\tan{\theta}}$

Problem 11:

Evaluate: $\lim_{x \rightarrow 0} \frac{3\sin {(x \deg)}- \sin{(3x \deg)}}{x^{3}}$

Problem 12:

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

Problem 13:

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

Problem 14:

Evaluate: $\lim_{x \rightarrow 0} \frac{5\sin{x}-7\sin{2x}+3\sin{3x}}{x^{2}\sin{x}}$

Problem 15:

Evaluate: $\lim_{x \rightarrow 0} \frac{x^{2}+1-\cos{x}}{x\tan{x}}$

Problem 16:

Evaluate: $\lim_{x \rightarrow \frac{\pi}{6}} \frac{\cos{x} - \sqrt{3}\sin{x}}{\pi - 6x}$

Problem 17:

Evaluate: $\lim_{x \rightarrow a} \frac{\sin{(\sqrt{x})}-sin{(\sqrt{a})}}{x-a}$

Problem 18:

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

Problem 19:

Evaluate: $\lim_{x \rightarrow 0}(1+\sin{x})^{\frac{1}{x}}$

Problem 20:

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

Problem 21:

Evaluate: $\lim_{x \rightarrow 1} x^{\frac{1}{x-1}}$

Problem 22:

Evaluate: $\lim_{x \rightarrow 0} (1+x+\frac{x^{2}}{4})^{\frac{1}{x}}$

Problem 23:

Evaluate: $\lim_{x \rightarrow 0} (\tan{(\frac{\pi}{4}+x)})^{\frac{1}{x}}$

Problem 24:

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

Problem 25:

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

Problem 26:

Evaluate: $\lim_{x \rightarrow 0} \frac{12^{x}-4^{x}-3^{x}+1}{x \tan{x}}$

Problem 27:

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

Problem 28:

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

Problem 29:

Evaluate: $\lim_{x \rightarrow 0} \frac{a^{x}+b^{x}+c^{x}-3^{(x+1)}}{\sin{x}}$

Problem 30:

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

Problem 31:

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

Problem 32:

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

Problem 33:

Evaluate: $\lim_{x \rightarrow 0} \frac{9^{x}-2 \times 3^{x}+1}{1-\cos{x}}$

Problem 34:

Evaluate: $\lim_{x \rightarrow 0} \frac{10^{x}+7^{x}-14^{x}-5^{x}}{x^{2}}$

Problem 35:

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

Problem 36:

Evaluate: $\lim_{x \rightarrow 1} \frac{ab^{x}-ba^{x}}{(x-1)}$

Problem 37:

Evaluate: $\lim_{x \rightarrow 2} \frac{ax^{2}-b}{x-2} = 4$ . Then, (i) $a=1, b=4$ (ii) $a=4, b=1$ (iii) $a=-4, b=1$ (iv) $a=2, b=1$

Problem 38:

Evaluate: $\lim_{x \rightarrow 2}\frac{x^{4}-8x}{\sqrt{x^{2}+21}-5}$

Problem 39:

Evaluate: $\lim_{x \rightarrow 2a} \frac{\sqrt{x-2a}+\sqrt{x} -\sqrt{2a}}{\sqrt{x^{2}-4a^{2}}}$

Problem 40:

Evaluate: $\lim_{x \rightarrow 4} \frac{x^{3}-64}{x^{3}-15x-4}$

Problem 41:

Evaluate: $\lim_{x \rightarrow 3} \frac{x^{2}+\sqrt{x+6}-12}{x^{2}-9}$

Problem 42:

Evaluate: $\lim_{x \rightarrow 2} \frac{x^{3}+\sqrt{x+2}-10}{x^{2}-4}$

Problem 43:

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

Problem 44:

Evaluate: $\lim_{x \rightarrow \infty} \sqrt{x} (\sqrt{x+2}-\sqrt{x})$

Problem 45:

Evaluate: $\lim_{h \rightarrow 0} \frac{h}{(a+h)^{8}-a^{8}}$

Problem 46:

Evaluate: $\lim_{x \rightarrow 1} \frac{x^{4}+x^{7}-2}{x^{3}-2x+1}$

Problem 47:

Evaluate: $\lim_{x \rightarrow 3} \frac{x+x^{2}+x^{3}-39}{x-3}$

Problem 48:

Evaluate: $\lim_{x \rightarrow \frac{\pi}{4}} \frac{2- cosec (x) * cosec(x)}{\cot{x}-1}$

Problem 49:

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

Problem 50:

Evaluate: $\lim_{x \rightarrow 0} \frac{\cos{8x} - \cos{2x}}{\cos{12x}-\cos{4x}}$

Regards,

Nalin Pithwa

Mathematics Hothouse