Limits Continuity And Differentiability - Calculus - Mathematics Previous Year Questions

$$The\,\,value\,\,of\,\,\mathop {\lim }\limits_{\theta \to 0} {{1 - \cos 4\theta } \over {1 - \cos 6\theta }}\,\,is$$

If $f(x)=\left\{\begin{array}{cl}\frac{\log _e x}{x-1} & ; x \neq 1 \\ k & ; x=1\end{array}\right.$

is continuous at $x=1$, then the value of $k$ is

If $f(x)=\left\{\begin{array}{clc}\frac{\sqrt{1+k x}-\sqrt{1-k x}}{x} & \text { if }-1 \leq x<0 \\ \frac{2 x+1}{x-1} & \text { if } 0 \leq x \leq 1\end{array}\right.$

is continuous at $x=0$, then the value of $k$ is

Rolle's theorem is not applicable in which one of the following cases?

\(\sum_\limits{r=1}^n(2 r-1)=x\) then, \(\lim _\limits{n \rightarrow \infty}\left[\frac{1^3}{x^2}+\frac{2^3}{x^2}+\frac{3^3}{x^2}+\ldots+\frac{n^3}{x^2}\right]=\)

If \(f(x)=\left\{\begin{array}{cl}\frac{\sin 3 x}{e^{2 x}-1} ; & x \neq 0 \\ k-2 ; & x=0\end{array}\right.\) is continuous at \(x=0\), then \(k=\)

The right hand and left hand limit of the function are respectively.

$$f(x)=\left\{\begin{array}{cc} \[\frac{e^{1 / x}-1}{e^{1 / x}+1}, & \text { if } x \neq 0 \\\] 0, & \text { if } x=0 \end{array}\right.$$

$$\lim _\limits{x \rightarrow 0}\left(\frac{\tan x}{\sqrt{2 x+4}-2}\right) \text { is equal to }$$

If \(f(x)=\left\{\begin{array}{cc}\frac{1-\cos K x}{x \sin x}, & \text { if } x \neq 0 \\ \frac{1}{2}, & \text { if } x=0\end{array}\right.\) is continuous at \(x=0\), then the value of \(K\) is

Consider the following statements

Statement 1 : \(\lim _\limits{x \rightarrow 1} \frac{a x^2+b x+c}{x^2+b x+a}\) is 1

(where \(a+b+c \neq 0\)).

Statement 2 : \(\lim _\limits{x \rightarrow -2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}\) is \(\frac{1}{4}\).

If \(f(x)=\left|\begin{array}{ccc}\cos x & 1 & 0 \\ 0 & 2 \cos x & 3 \\ 0 & 1 & 2 \cos x\end{array}\right|\), then \(\lim _\limits{x \rightarrow \pi} f(x)\) is equal to

At \(x=1\), the function

$$f(x)=\left\{\begin{array}{cc} x^3-1, & 1< x < \infty \\ x-1, & -\infty< x \leq 1 \end{array}\right. \text { is }$$

If \(f(x)=\left\{\begin{array}{cc}x^2-1, & 0< x<2 \\ 2 x+3, & 2 \leq x<3\end{array}\right.\),

the quadratic equation whose roots are \(\lim _\limits{x \rightarrow 2^{-}} f(x)\) and \(\lim _\limits{x \rightarrow 2^{+}} f(x)\) is

$$\lim _\limits{y \rightarrow 0} \frac{\sqrt{3+y^3}-\sqrt{3}}{y^3}=$$

If \(\lim _\limits{x \rightarrow 0} \frac{\sin (2+x)-\sin (2-x)}{x}=A \cos B\), then the values of \(A\) and \(B\) respectively are

The function \(f(x)=\cot x\) is discontinuous on every point of the set

$\lim \limits_{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} \cos x-1}{\cot x-1}$ is equal to

The function $f(x)=|\cos x|$ is