## Limits and Continuity: part 5: IITJEE Math: Tutorial problems for practice

Problem 1:

Find the value of the following limit:

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

Problem 2:

Find the values of the constant a and b such that the following limit is zero:

$\lim_{x \rightarrow \infty} [\frac{x^{2}+1}{x+1} -ax-b]$

Problem 3:

Find the value of the following limit:

$\lim_{\alpha \rightarrow \beta} \frac{\sin^{2}{\alpha}-\sin^{2}{\beta}}{\alpha^{2}-\beta^{2}}$

Problem 4:

If a, b, c, d are positive, then find the value of the following limit:

$\lim_{x \rightarrow \infty}(1+\frac{1}{a+bx})^{c+dx}$

Problem 5:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{(1-\cos{(2x)})\sin{(5x)}}{x^{2}\sin{(3x)}}$

Problem 6:

Find the value of the following limit:

$\lim_{x \rightarrow \infty} \frac{\sqrt{x^{2}-1}}{2x+1}$

Problem 7:

Find the value of the following limit:

$\frac{\log{(1+x+x^{2})}+\log{(1-x+x^{2})}}{\sec{x}-\cos{x}}$

Problem 8:

Find the value of the following limit:

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

Problem 9:

Find the value of f(0) such that the following function is continuous at zero:

$f(x) = (x+1)^{\cot{x}}$

Problem 10:

Let $f^{''}(x)$ be continuous at zero and $f^{''}(0)=4$. Then, find the numerical value of the following limit:

$\lim_{x \rightarrow 0}\frac{2f(x)-3f(2x)+f(4x)}{x^{2}}$

Problem 11:

Find the value of the following limit:

$\lim_{n \rightarrow \infty} (\frac{n^{3}}{3n^{2}-4} - \frac{n^{2}}{3n+2})$

Problem 12:

Find the values of x where the following function is discontinuous:

$f(x) = \frac{\sin{x} \log{(x-2)}}{(x^{2}-4x+3)}$

Problem 13:

The value of p for which the following function may be continuous at zero is what:

$f(x) = \frac{(4x-1)^{3}}{(\sin{\frac{x}{p}})(\log{(1+\frac{x^{2}}{3})})}$, when $x \neq 0$, and

$f(x) = 12(\log{4})^{3}$, when $x = 0$.

Problem 14:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{1-\cos{(mx)}}{1-\cos{(nx)}}$

Problem 15:

If $f(x) = \frac{4-7x}{3x+4}$ and $\lim_{x \rightarrow 2}f(x) = k$, and $\lim_{x \rightarrow 0}f(x) = m$, then the equation whose roots are $\frac{1}{k}, \frac{1}{m}$ is (a) $x^{2}+x=0$ (b) $x^{2}-1=0$ (c) $x^{2}+1=0$ (d) $x^{2}+2x=0$

Problem 16:

Find the value of the following limit:

$\lim_{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots + x^{n}-n}{x-1}$

Problem 17:

Find the value of the following limit:

$\lim_{x \rightarrow 1} \frac{\sqrt[n]{x^{m}}-1}{\sqrt[m]{x^{n}}-1}$

Problem 18:

Find the value of the following limit:

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

Regards,

Nalin Pithwa

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