## Limits and Continuity: Part 4: IITJEE Math Tutorial Problems for Practice

Problem 1:

If $\alpha, \beta$ are the two roots of the quadratic equation $ax^{2}+bx+c=0$, then the find the value of the following limit:

$\lim_{x \rightarrow \alpha} \frac{1-\cos{(ax^{2}+bx+c)}}{(x-\alpha^{2})}$

Problem 2: Given the following functio; find the value of f(0) so that the function is continuous at zero:

$f(x) = \frac{\sqrt{1+x}-(1+x)^{\frac{1}{3}}}{x}$ when $x \neq 0$.

Problem 3:

Find the value of the following limit: $\lim_{x \rightarrow 0} \frac{\sin {(x \deg)}}{x}$

Problem 4:

If $\lim_{x \rightarrow 0}\frac{1-\cos{(1-\cos{x})}}{x^{4}}=k$, which numerical value divides $k^{-2}$?

Problem 5:

Find the value of the following limit:

$\lim_{x \rightarrow 1} {(\sec{(\frac{\pi x}{2})})(\log{x})}$

Problem 6:

Find the value of the following limit:

$\lim_{x \rightarrow \infty} \frac{(2+x)^{40}(4+x)^{5}}{(2-x)^{45}}$

Problem 7:

Let it be given that $L = \lim_{x \rightarrow 2} (x^{3}-x^{2}+x-1)$ and $M = \lim_{x \rightarrow -2}(x^{4}-x^{3}+x^{2}-x)$ then find the value of the following limit:

$\lim_{x \rightarrow 1}\frac{Lx^{2}-Mx+2}{Mx^{2}-Lx-2}$

Problem 8:

Find the value of the following limit:

$\lim_{x \rightarrow \arctan{-3}} \frac{\tan^{2}{x}-2\tan{x}-3}{\tan^{2}-4\tan{x}+3}$

Problem 9:

Find the value of the following limit:

$\lim_{x \rightarrow \infty} (\frac{x+6}{x+1})^{x+4}$

Problem 10:

Find the value of b such that the following function is continuous at every point of its domain:

$f(x) = 5x-4$, when $0 < x \leq 1$, and $f(x)=4x^{2}+3bx$, when $1 < x < 2$.

Problem 11:

Find the value of the following limit:

$\lim_{x \rightarrow 0}(\cos{x})^{\frac{1}{x}}$

Problem 12:

Find the value of the following limit:

$\lim_{h \rightarrow 0} \frac{\tan{(x+h)-\tan{x}}}{x}$

Problem 13:

Consider the following: $\lim_{x \rightarrow 0}\frac{\sqrt{1-\cos{2x}}}{x}$. Then, which of the following is true (a) limit exists and is equal to $\sqrt{2}$ (b) exists and it equals $-\sqrt{2}$ (c) limit does not exist because $x - 1 \rightarrow 0$ (d) limit does not exist because left hand limit is not equal to right hand limit

Problem 14:

Find the value of the following limit:

$\lim_{x \rightarrow \frac{\pi}{4}} (\frac{1-\tan{x}}{1-\sqrt{2}\sin{x}})$

Problem 15:

Find the value of p given the following:

$\lim_{x \rightarrow 0} \frac{\sin{px}}{\tan{3x}}=4$

Problem 16:

The number of points of discontinuity of the function $f(x)= \frac{1}{\log{|x|}}$ is (a) zero (b) 1 (c) 2 (d) 3

Problem 17:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{\sin(\pi \cos^{2}{x})}{x^{2}}$

Problem 18:

Find the value of the following limit:

$\lim_{x \rightarrow 2}\frac{3^{\frac{x}{2}}-3}{3^{x}-9}$

Regards,

Nalin Pithwa

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