Limits Continuity And Differentiability - Calculus - Mathematics Previous Year Questions

If \(f(x-y)=f(x) \circ g(y)-f(y) \circ g(x)\) And \(g(x-y) =g(x) \circ g(y)+f(x) \circ f(y)\) for all \(x, y \in \mathrm{R}\). If right-hand derivative at \(x=0\) exists for \(f(x)\), find the derivative of \(g(x)\) at \(x=0\)

If $f(x)=\min \left\{1, x^2, x^3\right\}$, then

For $x>0, \mathop {\lim }\limits_{x \to 0}\left((\sin x)^{1 / x}+(1 / x)^{\sin x}\right)$ is :

Let \(f(x)=2+\cos x\) for all real \(x\).

STATEMENT - 1 : For each real \(t\), there exists a point \(c\) in \([t, t+\pi]\) such that \(f^{\prime}(C)=0\).

STATEMENT - 2 : \(f(t)=f(t+2 \pi)\) for each real \(t\).

For \(k > 0\), the set of all values of \(k\) for which \(k e^{x}-x=0\) has two distinct roots is

The line \(y=x\) meets \(y=k e^{\mathrm{x}}\) for \(k \leq 0\) at

The positive value of \(k\) for which \(k e^{x}-x=0\) has only one root is

Let \(g(x) = {{{{(x - 1)}^n}} \over {\log {{\cos }^m}(x - 1)}};0 < x < 2,m\) and \(n\) are integers, \(m \ne 0,n > 0\), and let \(p\) be the left hand derivative of \(|x - 1|\) at \(x = 1\). If \(\mathop {\lim }\limits_{x \to {1^ + }} g(x) = p\), then

Which of the following is true?

Let \(L = \mathop {\lim }\limits_{x \to 0} {{a - \sqrt {{a^2} - {x^2}} - {{{x^2}} \over 4}} \over {{x^4}}},a > 0\). If L is finite, then

Let f : R \(\to\) R be a function such that \(f(x + y) = f(x) + f(y),\,\forall x,y \in R\). If f(x) is differentiable at x = 0, then

If \(f(x) = \left\{ {\matrix{ \[{ - x - {\pi \over 2},} & {x \le - {\pi \over 2}} \cr\] \[{ - \cos x} & { - {\pi \over 2} < x \le 0} \cr\] {x - 1} & {0 < x \le 1} \cr {\ln x} & {x > 1} \cr } } \right.\), then

If \(\mathop {\lim }\limits_{x \to 0} {[1 + x\ln (1 + {b^2})]^{1/x}} = 2b{\sin ^2}\theta\), \(b > 0\) and \(\theta \in ( - \pi ,\pi ]\), then the value of \(\theta\) is

If \(\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4\), then

Let \(f(x) = \left\{ {\matrix{ \[{{x^2}\left| {\cos {\pi \over x}} \right|,} & {x \ne 0} \cr\] {0,} & {x = 0} \cr } } \right.\)

x\(\in\)R, then f is

For every integer n, let a_n and b_n be real numbers. Let function f : R \(\to\) R be given by

\(f(x) = \left\{ {\matrix{ \[{{a_n} + \sin \pi x,} & {for\,x \in [2n,2n + 1]} \cr\] \[{{b_n} + \cos \pi x,} & {for\,x \in (2n - 1,2n)} \cr\] } } \right.\), for all integers n. If f is continuous, then which of the following hold(s) for all n ?

\(a \in R\) (the set of all real numbers), a \(\ne\) \(-\)1,

\(\mathop {\lim }\limits_{n \to \infty } {{({1^a} + {2^a} + ... + {n^a})} \over {{{(n + 1)}^{a - 1}}[(na + 1) + (na + 2) + ... + (na + n)]}} = {1 \over {60}}\), Then a = ?