Limits Continuity And Differentiability - Calculus - Mathematics Previous Year Questions

The value of \(\lim _\limits{n \rightarrow \infty}\left\{\frac{1+2+3+\ldots+n}{n+2}-\frac{n}{2}\right\}\) is

\(\lim _\limits{x \rightarrow 0}\left\{\tan \left(\frac{\pi}{4}+x\right)\right\}^{1 / x}\) is equal to

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

The value of \(\lim _\limits{t \rightarrow \infty} \frac{\ln \left(\frac{3}{2} t\right)}{t^2}\)

The value of \(f(x) = \mathop {\lim }\limits_{x \to 2} {{{x^3} - 3{x^2} + 4} \over {{x^4} - 7x - 2}}\)

If \(f(x)=\left\{\begin{array}{cc}(\sin x+\cos x)^{\operatorname{cosec} x} & ,-\frac{\pi}{2}< x<0 \\ a & ,x=0 \\ \frac{e^{1 / x}+e^{2 / x}+e^{3 / x}}{a e^{-2+\frac{1}{x}}+b e^{-1+\frac{3}{x}}} & , 0< x<\frac{\pi}{2}\end{array}\right.\) is continuous at \(x=0\), then the value of \((b, a)\) is

The value of \(\lim _\limits{x \rightarrow \infty}\left[\frac{p^{1 / x}+q^{1 / x}+r^{1 / x}+s^{1 / x}}{4}\right]^{3 x}, p, q, r, s>0 \text {, }\) is

For \(x \in R, f(x)=|\log 2-\sin x|\) and \(g(x)=f(f(x))\), then

$\lim \limits_{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2}$ equal to

$\lim _\limits{x \rightarrow 0} \frac{1}{x} \int_0^x(1+\sin 3 t)^{\frac{1}{t}} d t$ is equal to