## Limits and Continuity: Part 3: IITJEE maths tutorial problems

Problem 1:

Find the following limit:

$\lim_{h \rightarrow 0} 2 \times \frac{\sqrt{3}(\sin{(\frac{\pi}{6}+h)})-\cos{(\frac{\pi}{6}+h)}}{\sqrt{3}h(\sqrt{3}\cos{h}-\sin{h})}$

Problem 2:

Let the given function be continuous in the interval $[-1,1]$. Then what must be the value of p?

$f(x) = \frac{\sqrt{(1+px)}-\sqrt{(1-px)}}{x}$, when $-1 \leq x \leq 0$

$f(x) = \frac{2x+1}{x-2}$, when $0 \leq x \leq 1$.

Problem 3:

Let the given function be continuous for $0 \leq x < \infty$, then find the most suitable values for a and b:

$f(x) = \frac{x^{2}}{a}$, for $0 \leq x <1$

$f(x) = a$, for $1 \leq x < \sqrt{2}$

$f(x) = \frac{2b^{2}-4b}{x^{2}}$, for $\sqrt{2} \leq x < \infty$

Problem 4:

Find the value of the following:

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

Problem 5:

The function $f(x) = \frac{1}{x} \times (\sqrt{(1+\sin{x})} - \sqrt{(1-\sin{x})})$ is not defined at $x=0$. The value of $f(0)$ so that f(x) becomes continuous at $x=0$ is (a) 1 (b) 2 (c) 0 (d) none

Problem 6:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{a^{x}-1}{\sqrt{(1+x)}-1}$

Problem 7:

Let the given function be $f(x) = \frac{\tan{(\frac{\pi}{4}-x)}}{\cot{(2x)}}$. Find the value which should be assigned to f at $x = \frac{\pi}{4}$ so that f is continuous everywhere on the reals.

Problem 8:

Let it be given that $n \in N$ and $f(x) = \frac{1+2^{x}+3^{x}+\ldots + n^{x}-n}{x}$, x is not zero. What value of f(0) will make the function f continuous on the reals.

Problem 9:

Find the value of the following limit:

$\lim_{\theta \rightarrow 0^{+}}\frac{\sin{\sqrt{\theta}}}{\sqrt{(\sin{\theta})}}$

Problem 10:

If $a = \log_{3}{(3x)}$ and $b = \log_{x}{(3)}$, then the find the limiting value of $a^{b}$ as $x \rightarrow 1$:

Problem 11:

Let it be given that $n \in N$. Then, the find the value of the following limit:

$\lim_{x \rightarrow 0}\frac{\sin{x}+\sin{(2x)}+\ldots + \sin{(nx)}}{\sin{x}+\sin{(3x)}+\sin{(5x)}+\ldots + \sin{(2n-1)x}}$

Problem 12:

Let it be given that $f(x) = x \sin{(\frac{1}{x})}$ when x is not zero and $f(x) = 0$, when x is zero. Then, find the value of the following limit:

$\lim_{x \rightarrow 0}f(x)$.

Problem 13:

Find the value of the following limit:

$\lim_{x \rightarrow 0}\frac{e^{x^{2}}-\cos{(x)}}{x^{2}}$

Problem 14:

Let it be given that $f(x) = \frac{x^{2}-(A+2)x+A}{x-2}$ when $x \neq 2$ and $f(x) = 2$, when $x=2$ is continuous at $x=2$. Then, find the value of A.

Regards,

Nalin Pithwa

This site uses Akismet to reduce spam. Learn how your comment data is processed.