Limits and continuity: part 6: IITJEE math: Tutorial Problems for Practice

Problem 1:

Find the value of the following limit:

$\lim_{y \rightarrow x} \frac{y^{y}-x^{x}}{y-x}$

Problem 2:

Find the value of the following limit:

$\lim_{x \rightarrow 2} \frac{e^{\log {(\frac{3x+4}{x})}} - e^{\log{5}}}{x-2}$

Problem 3:

Find the value of the following limit:

$\lim_{h \rightarrow 0} \frac{(a+h)^{2} \sin{(a+h)}-a^{2}\sin{a}}{h}$. Choose one of the following: (i) $a\cos{a} -2 \sin{a}$ (ii) $a \cos{a} +2\sin{a}$ (iii) a(a\cos{a} + 2 \sin{a}) (iv) $a^{2} \cos{a}$

Problem 4:

Find the value of the following limit:

$\lim_{x \rightarrow -2} \frac{x^{5}+2x^{4}+x^{2}+3x+2}{x^{4}+2x^{3}+3x^{2}-5x-22}$

Problem 5:

Find the value of the following limit:

$\lim_{x \rightarrow \infty} (\frac{x+1}{x+2})(\frac{2x+1}{3x+4})$

Problem 6:

Find the value of the following limit:

$\lim_{x \rightarrow \infty} \frac{(x+1)^{10}+(x+2)^{10} + \ldots + (x+100)^{10}}{(x^{10}+10^{10})}$

Problem 7:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{\cos{(ax)}-\cos{(bx)}}{\cos{(cx)} - \cos{(dx)}}$

Problem 8:

Find the value of the following limit:

$\lim_{x \rightarrow \pi} \frac{1-\cos{(7(x-\pi))}}{5(x-\pi)^{2}}$

Problem 9:

Find the value of the following limit:

$\lim_{x \rightarrow 1} \frac{\sqrt[3]{x^{2})-2.\sqrt[3]{(x)}+1}()}{(x-1)^{2}}$

Problem 10:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{\sec{(4x)} - \sec{(2x)}}{(\sec{(3x)}-\sec{x})}$

Problem 11:

Find the value of the following limit:

$\lim_{x \rightarrow 1} \frac{1+\cos{\pi x}}{\tan^{2}{\pi x}}$

Problem 12:

Find the value of the following limit:

$\lim_{x \rightarrow 1} \frac{3 \sin{x\pi } -\sin{3x \pi}}{(x-1)^{3}}$

Problem 13:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{\tan^{4}{x} - \sin^{4}{x}}{x^{6}}$

Problem 14:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{x^{3}\sin{x}}{(\sec{x} - \cos{x})^{2}}$

Problem 15:

If the value of the following limit is -1, then find the value of a:

$\lim_{x \rightarrow a} \frac{\sin{x} -\sin{a}}{\cos{x} -\cos{a}}$

Problem 16:

Find the value of the following limit:

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

Problem 17:

Find $\lim_{x \rightarrow 0} \frac{f^{'}(x)}{x}$ if f(x) is given as follows:

$f(x) = \left | \begin{array}{ccc} \cos{x} & x & 1 \\ 2\sin{x} & x^{2} & 2x \\ \tan{x} & x & 1 \end{array} \right |$

Problem 18:

If $f(a) = \lim_{x \rightarrow \infty} x(a^{\frac{1}{x}}-1)$, then $f(ab)$ is equal to (i) $f(a).f(b)$ (ii) $f(a)+f(b)$ (iii) $0$ (iv) ab

Problem 19:

Evaluate the following limit:

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

Problem 20:

The function f is defined by :

$f(x) = \frac{e^{x}+e^{-x}-2}{x \sin{x}}$ in the interval $[\frac{\pi}{2}, - \frac{\pi}{2}]-\{0\}$

In order for this function to be continuous in $[\frac{\pi}{2}, -\frac{\pi}{2}]$, we have to define (a) $f(0)=2$ (b) $f(\frac{\pi}{2})=1$ (c) $f(0)=1$ (d) $f(-\frac{\pi}{2})=1$

Problem 21:

The function $f(x) = \frac{|x|}{x}$ when $x \neq 0$ and $f(0)=0$, (a) has removable discontinuity at $x=0$ (b) has irremovable discontinuity at $x=0$ (c) is continuous at $x=0$ (d) $\lim_{x \rightarrow 0}f(x)$ exists.

Problem 22:

Let $f(x)$ be defined by

$f(x) = \sin{2x}$ if $0 < x \leq \frac{\pi}{6}$

$f(x) = ax + b$, if $\frac{\pi}{6} < x \leq 1$

If $f(x)$ and $f^{'}(x)$ are continuous in $(0,1)$, then the value of b is (i) $\frac{1}{2} - \frac{\pi}{6}$ (ii) $\frac{1}{2} - \frac{\pi}{3}$ (iii) $\frac{\sqrt{3}}{2} + \frac{\pi}{6}$ (iv) $\frac{\sqrt{3}}{2} - \frac{\pi}{6}$

Problem 23:

If $f(x) = x^{\frac{2}{3}}-2$, where $x \geq 0$, then $\lim_{x \rightarrow 2} f^{-1}(x)$ is (a) 7 (b) 8 (c) 9 (d) 10

Problem 24:

If $f(x) = \frac{x+1}{x+2}$ and $g(x) = \frac{1}{x}$ then $\lim_{x \rightarrow 2} (f+g)(x)$ is (a) $\frac{1}{5}$ (b) $\frac{2}{5}$ (c) $\frac{3}{5}$ (d) $\frac{4}{5}$

Regards,

Nalin Pithwa

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