Limits, Continuity and Differentiability

Let \(f(x) = 3{x^{10}} - 7{x^8} + 5{x^6} - 21{x^3} + 3{x^2} - 7\). Then \(\mathop {\lim }\limits_{h \to 0} {{f(1 - h) - f(1)} \over {{h^3} + 3h}}\)

Let for all x > 0, \(f(x) = \mathop {\lim }\limits_{n \to \infty } n({x^{1/n}} - 1)\), then

Consider the non-constant differentiable function f one one variable which obeys the relation \({{f(x)} \over {f(y)}} = f(x - y)\). If f' (0) = p and f' (5) = q, then f' (\(-\)5) is

If f'' (0) = k, k \(\ne\) 0, then the value of \(\mathop {\lim }\limits_{x \to 0} {{2f(x) - 3f(2x) + f(4x)} \over {{x^2}}}\) is

Let f : R \(\to\) R be twice continuously differentiable. Let f(0) = f(1) = f'(0) = 0. Then,

\(\mathop {\lim }\limits_{x \to 0} {(\sin x)^{2\tan x}}\) is equal to

Let \(f(x) = \left\{ {\matrix{ {{{{x^p}} \over {{{(\sin x)}^q}}},} & {if\,0 < x \le {\pi \over 2}} \cr {0,} & {if\,x = 0} \cr } } \right.\), \((p,q \in R)\). Then, Lagrange's mean value theorem is applicable to f(x) in closed i...

If \(y = (1 + x)(1 + {x^2})(1 + {x^4})...(1 + {x^{2n}})\), then the value of \(\left( {{{dy} \over {dx}}} \right)\) at x = 0 is

The value of \(\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\sqrt {n + 1} + \sqrt {n + 2} + ... + \sqrt {2n - 1} } \over {{n^{3/2}}}}} \right\}\) is

\(\mathop {\lim }\limits_{x \to 1} {\left( {{{1 + x} \over {2 + x}}} \right)^{{{(1 - \sqrt x )} \over {(1 - x)}}}}\) is equal to

Let R be the set of real numbers and f : R \(\to\) R be such that for all x, y \(\in\) R, \(|f(x) - f(y)| \le |x - y{|^3}\). Prove that f is a constant function.

The function f(x) = ax + b is strictly increasing for all real x if

\(f(x) = \left\{ {\matrix{ {0,} & {x = 0} \cr {x - 3,} & {x > 0} \cr } } \right.\)
The function f(x) is

For the function \(f(x) = {e^{\cos x}}\), Rolle's theorem is

\(f(x) = \left\{ {\matrix{ {[x] + [ - x],} & {when\,x \ne 2} \cr {\lambda ,} & {when\,x = 0} \cr } } \right.\)
If f(x) is continuous at x = 2, the value of \(\lambda\) will be

If the function \(f(x) = \left\{ {\matrix{ {{{{x^2} - (A + 2)x + A} \over {x - 2}},} & {for\,x \ne 2} \cr {2,} & {for\,x = 2} \cr } } \right.\) is continuous at x = 2, then

\(\mathop {\lim }\limits_{x \to 0} {{\sin (\pi {{\sin }^2}x)} \over {{x^2}}} =\)

The value of \(\mathop {\lim }\limits_{x \to 1} {{x + {x^2} + ..... + {x^n} - n} \over {x - 1}}\) is

If f(a) = 2, f'(a) = 1, g(a) = \(-\)1 and g'(a) = 2, find the value of \(\mathop {\lim }\limits_{x \to a} {{g(x)f(a) - g(a)f(x)} \over {x - a}}\).

Use the formula \(\mathop {\lim }\limits_{x \to 0} {{{a^x} - 1} \over x} = {\log _e}a\), to compute \(\mathop {\lim }\limits_{x \to 0} {{{2^x} - 1} \over {\sqrt {1 + x} - 1}}\).

If N = n! (n \(\in\) N, n > 2), then find \(\mathop {\lim }\limits_{N \to \infty } \left[ {{{({{\log }_2}N)}^{ - 1}} + {{({{\log }_3}N)}^{ - 1}} + \,\,.....\,\, + {{({{\log }_n}N)}^{ - 1}}} \right]\).

In which of the following functions, Rolle's theorem is applicable?

\(\mathop {\lim }\limits_{x \to 0} {{\sin |x|} \over x}\) is equal to

The value of f(0) so that the function \(f(x) = {{1 - \cos (1 - \cos x)} \over {{x^4}}}\) is continuous everywhere is

If \(y = (1 + x)(1 + {x^2})(1 + {x^4})\,.....\,(1 + {x^{2n}})\), then the value of \({\left( {{{dy} \over {dx}}} \right)_{x = 0}}\) is

If \(f(5) = 7\) and \(f'(5) = 7\), then \(\mathop {\lim }\limits_{x \to 5} {{x\,f(5) - 5f(x)} \over {x - 5}}\) is given by

The value of \(\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + 5{x^2}} \over {1 + 3{x^2}}}} \right)^{{1 \over {{x^2}}}}}\) is

The value of \(\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x + \cos x - 1} \over {{x^2}}}\) is

The value of \(\mathop {\lim }\limits_{x \to 1} {{\sin ({e^{x - 1}} - 1)} \over {\log x}}\) is

\(f(x) = x + |x|\) is continuous for

Let \(f(x) = {{\sqrt {x + 3} } \over {x + 1}}\), then the value of \(\mathop {\lim }\limits_{x \to - 3 - 0} f(x)\) is

A function f(x) is defined as follows for real x
\(f(x) = \left\{ {\matrix{ {1 - {x^2}} & , & {for\,x < 1} \cr 0 & , & {for\,x = 1} \cr {1 + {x^2}} & , & {for\,x > 1} \cr } } \right.\)
Then

The \(\mathop {\lim }\limits_{x \to 2} {5 \over {\sqrt 2 - \sqrt x }}\) is

The value of the limit \(\mathop {\lim }\limits_{x \to 2} {{{e^{3x - 6}} - 1} \over {\sin (2 - x)}}\) is

Rolle's theorem is not applicable to the function \(f(x) = |x|\) for \(- 2 \le x \le 2\) because

\(\mathop {\lim }\limits_{x \to {\pi \over 2}} {{{a^{\cot x}} - {a^{\cos x}}} \over {\cot x - \cos x}},a > 0\)