Limits Continuity And Differentiability - Calculus - Mathematics Previous Year Questions

Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix defined by $a_{i j}=\left\{\begin{array}{cc}k^i, & \forall i=j \\ 0, & \text { otherwise }\end{array}\right.$. If $m=$ trace of $A$ and $\lim _{k \rightarrow 1} \frac{n-m}{1-k}=171$, then the value of $n$ is

$$\mathop {\lim }\limits_{x \to \infty } {x^3}\left[\sqrt{x^2+\sqrt{x^4+1}}-\sqrt{2 x}\right]=$$

Let $f(x)=\left\{\begin{array}{ccc}3-x & \text { if } & x<-3 \\ 6 & \text { if } & -3 \leq x \leq 3 . \text { Let } \alpha \text { be the number } \\ 3+x & \text { if } & x>3\end{array}\right.$ of points of discontinuity of $f$ and $\beta$ be the number of points where $f$ is not differentiable. Then, $\alpha+\beta=$

If $f(x)=\left\{\begin{array}{l}\frac{x^2-16}{x-4} \text { if } x>4 \\ 2 x \quad \text { if } x \leq 4\end{array}\right.$ then $f^{\prime}\left(4^{-}\right)+f^{\prime}\left(4^{+}\right)=$

$$\lim _{x \rightarrow 0} \frac{2^{2 x}-2^{x+1}+2-\cos 2 x}{x^2}=$$

$$\lim _{x \rightarrow 3^{-}} \frac{x^3-3 x^2-4 x+12}{2 x^3-7 x^2+2 x+3}=$$

$$\lim\limits_{x \rightarrow 1}(1-x) \tan \left(\frac{\pi}{2} x\right)=$$

If $f(9)=9$ and $f^{\prime}(9)=4$, then $\lim\limits_{x \rightarrow 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3}=$

$\mathop {\lim }\limits_{x \to 0} \frac{\tan ^4 x-\sin ^4 x}{x^6}=$

$$\mathop {\lim }\limits_{x \to 2} \frac{\sqrt[3]{6+x}-\sqrt[3]{10-x}}{x-2}=$$

$$\mathop {\lim }\limits_{x \to 2 + }\left([x]^2-[x]-2\right)+\mathop {\lim }\limits_{x \to- 3 - }\left([x]^2-4[x]+3\right)=$$

$$\lim _{x \rightarrow 0} \frac{\left(3^{2 x}-\sqrt{x+1}\right) \sin 5 x}{1-\cos 4 x}=$$

If a function $f$ is defined by $f(x)=\frac{\cot ^3 x-\tan x}{\cos (x+\pi / 4)},(x \neq \pi / 4)$, then $\lim _{x \rightarrow \pi / 4} f(x)=$

$$\begin{array}{r} \lim _{x \rightarrow 0} \frac{2 \tan x+\cos x-1+x}{\sqrt{4 \sin ^2 x+2 \tan x+1}}= \\ -\sqrt{3 \tan ^2 x+\sin x+1} \end{array}$$

$$\lim _{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n\left(k^2 x\right)=$$

The quadratic equation whose roots are

$$l=\lim _{\theta \rightarrow 0}\left(\frac{3 \sin \theta-4 \sin ^3 \theta}{\theta}\right) \text { and } m=\lim _{\theta \rightarrow 0}\left(\frac{2 \tan \theta}{\theta\left(1-\tan ^2 \theta\right)}\right) \text { is }$$

The value of ' $a$ ' for which the function

$f(x)=\left\{\begin{array}{cl}\frac{1-\cos 4 x}{x^2}, & x<0 \\ \frac{a}{\sqrt{x}}, & x=0 \text { is continuous at } x=0, \text { is } \\ \frac{\sqrt{16+\sqrt{x}}-4}{\sqrt{16+}} & \end{array}\right.$

$$\mathop {\lim }\limits_{x \to 0} \frac{1-\cos \left(x^2+\pi(x+2)\right)}{x^2}=$$

If $\log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots \ldots \infty$ and $\mathop {\lim }\limits_{x \to 0} \frac{\log (1+x)^{1+x}}{x^2}-\frac{1}{x}=k$, then $12 k=$

If $f(x)=\left\{\begin{array}{ll}k, & \text { for } x=1 \\ \frac{(9 x-1)(\sqrt{x}-1)}{3 x^2+2 x-5}, & \text { for } x \neq 1\end{array}\right.$ is continuous on $[0, \infty)$, then $k=$