Limits, Continuity and Differentiability

$$ \text { Let the function } f(x)=\left\{\begin{array}{cl} \frac{\log _{e}(1+5 x)-\log _{e}(1+\alpha x)}{x} & ;\text { if } x \neq 0 \\ 10 & ; \text { if } x=0 \end{array} \text { be continuous at } x=0 .\right. $$
Then \(\alpha\) is equal...

Let f : R \(\to\) R be defined as
\(f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.\)
w...

If \(\mathop {\lim }\limits_{x \to 1} {{\sin (3{x^2} - 4x + 1) - {x^2} + 1} \over {2{x^3} - 7{x^2} + ax + b}} = - 2\), then the value of (a \(-\) b) is equal to ___________.

The value of \(\mathop {\lim }\limits_{n \to \infty } 6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {{r^2} + 3r + 3}}} \right)} } \right\}\) is equal to :

Let f, g : R \(\to\) R be functions defined by
\(f(x) = \left\{ {\matrix{ {[x]} & , & {x < 0} \cr {|1 - x|} & , & {x \ge 0} \cr } } \right.\) and $$g(x) = \left\{ {\matrix{
{{e^x} - x} & , & {x < 0} \cr
{{{(x - 1)}^2} - ...

\(\lim\limits_{x \rightarrow 0}\left(\frac{(x+2 \cos x)^{3}+2(x+2 \cos x)^{2}+3 \sin (x+2 \cos x)}{(x+2)^{3}+2(x+2)^{2}+3 \sin (x+2)}\right)^{\frac{100}{x}}\) is equal to ___________.

Let \(f:[0,1] \rightarrow \mathbf{R}\) be a twice differentiable function in \((0,1)\) such that \(f(0)=3\) and \(f(1)=5\). If the line \(y=2 x+3\) intersects the graph of \(f\) at only two distinct points in \((0,1)\), then the least numbe...

The function \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x)=\lim\limits_{n \rightarrow \infty} \frac{\cos (2 \pi x)-x^{2 n} \sin (x-1)}{1+x^{2 n+1}-x^{2 n}}\) is continuous for all x in :

Let a be an integer such that \(\mathop {\lim }\limits_{x \to 7} {{18 - [1 - x]} \over {[x - 3a]}}\) exists, where [t] is greatest integer \(\le\) t. Then a is equal to :

Let [t] denote the greatest integer \(\le\) t and {t} denote the fractional part of t. The integral value of \(\alpha\) for which the left hand limit of the function
$$f(x) = [1 + x] + {{{\alpha ^{2[x] + {\{x\}}}} + [x] - 1} \over {2[x] + \...

If for \(\mathrm{p} \neq \mathrm{q} \neq 0\), the function \(f(x)=\frac{\sqrt[7]{\mathrm{p}(729+x)}-3}{\sqrt[3]{729+\mathrm{q} x}-9}\) is continuous at \(x=0\), then :

Let f, g : R \(\to\) R be two real valued functions defined as \(f(x) = \left\{ {\matrix{ { - |x + 3|} & , & {x < 0} \cr {{e^x}} & , & {x \ge 0} \cr } } \right.\) and $$g(x) = \left\{ {\matrix{
{{x^2} + {k_1}x} & , & {x < 0} ...

\(\mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x)}}\) is equal to :

Let f(x) = min {1, 1 + x sin x}, 0 \(\le\) x \(\le\) 2\(\pi\). If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to

\(\mathop {\lim }\limits_{x \to 0} {{\cos (\sin x) - \cos x} \over {{x^4}}}\) is equal to :

Let \(f(x) = \left\{ {\matrix{ {{x^3} - {x^2} + 10x - 7,} & {x \le 1} \cr { - 2x + {{\log }_2}({b^2} - 4),} & {x > 1} \cr } } \right.\).
Then the set of all values of b, for which f(x) has maximum value at x = 1, is :

If \(f(x) = \left\{ {\matrix{ {x + a} & , & {x \le 0} \cr {|x - 4|} & , & {x > 0} \cr } } \right.\) and \(g(x) = \left\{ {\matrix{ {x + 1} & , & {x < 0} \cr {{{(x - 4)}^2} + b} & , & {x \ge 0} \cr } } \right.\) are c...

If the function $$f(x) = \left\{ {\matrix{
{{{{{\log }_e}(1 - x + {x^2}) + {{\log }_e}(1 + x + {x^2})} \over {\sec x - \cos x}}} & , & {x \in \left( {{{ - \pi } \over 2},{\pi \over 2}} \right) - \{ 0\} } \cr
k & , & {x = 0} \cr

...

Let f : R \(\to\) R be a continuous function such that \(f(3x) - f(x) = x\). If \(f(8) = 7\), then \(f(14)\) is equal to :

Let \(\beta=\mathop {\lim }\limits_{x \to 0} \frac{\alpha x-\left(e^{3 x}-1\right)}{\alpha x\left(e^{3 x}-1\right)}\) for some \(\alpha \in \mathbb{R}\). Then the value of \(\alpha+\beta\) is :

Let f(x) be a polynomial function such that \(f(x) + f'(x) + f''(x) = {x^5} + 64\). Then, the value of \(\mathop {\lim }\limits_{x \to 1} {{f(x)} \over {x - 1}}\) is equal to:

Let \(f(x) = \left[ {2{x^2} + 1} \right]\) and \(g(x) = \left\{ {\matrix{ {2x - 3,} & {x < 0} \cr {2x + 3,} & {x \ge 0} \cr } } \right.\), where [t] is the greatest integer \(\le\) t. Then, in the open interval (\(-\)1, 1), the ...

\(\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{{\tan }^2}x\left( {{{(2{{\sin }^2}x + 3\sin x + 4)}^{{1 \over 2}}} - {{({{\sin }^2}x + 6\sin x + 2)}^{{1 \over 2}}}} \right)} \right)\) is equal to

Let $$f(x)=\left\{\begin{array}{l}\left|4 x^{2}-8 x+5\right|, \text { if } 8 x^{2}-6 x+1 \geqslant 0 \\ {\left[4 x^{2}-8 x+5\right], \text { if } 8 x^{2}-6 x+1<0,}\end{array}\right.$$ where \([\alpha]\) denotes the greatest integer less tha...

If \(\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} - n - 1} + n\alpha + \beta } \right) = 0\), then \(8(\alpha+\beta)\) is equal to :

\(\lim\limits_{x \rightarrow \frac{\pi}{4}} \frac{8 \sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2} \sin 2 x}\) is equal to

The number of points where the function
\(f(x) = \left\{ {\matrix{ {|2{x^2} - 3x - 7|} & {if} & {x \le - 1} \cr {[4{x^2} - 1]} & {if} & { - 1 < x < 1} \cr {|x + 1| + |x - 2|} & {if} & {x \ge 1} \cr } } \right.\)
[t] denote...

Let \(f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|x|]\} } & {,\,|x| < 1} \cr 1 & {,\,otherwise} \cr } } \right.\)
where [t] denotes greatest integer \(\le\) t. I...

\(\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}\left( {\pi {{\cos }^4}x} \right)} \over {{x^4}}}\) is equal to :

If the function $$f(x) = \left\{ {\matrix{
{{1 \over x}{{\log }_e}\left( {{{1 + {x \over a}} \over {1 - {x \over b}}}} \right)} & , & {x < 0} \cr
k & , & {x = 0} \cr
{{{{{\cos }^2}x - {{\sin }^2}x - 1} \over {\sqrt {{x^2} + 1} ...

The function \(f(x) = \left| {{x^2} - 2x - 3} \right|\,.\,{e^{\left| {9{x^2} - 12x + 4} \right|}}\) is not differentiable at exactly :

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then

If \(\alpha = \mathop {\lim }\limits_{x \to {\pi \over 4}} {{{{\tan }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}\) and \(\beta = \mathop {\lim }\limits_{x \to 0 } {(\cos x)^{\cot x}}\) are the roots of the equation, ...

Let \(f:[0,3] \to R\) be defined by \(f(x) = \min \{ x - [x],1 + [x] - x\}\) where [x] is the greatest integer less than or equal to x. Let P denote the set containing all x \(\in\) [0, 3] where f i discontinuous, and Q denote the set cont...

Let f : R \(\to\) R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of \(\mathop {\lim }\limits_{x \to 2} {{{x^2}f(2) - 4f(x)} \over {x - 2}}\) is equal to :

Let \(f:\left( { - {\pi \over 4},{\pi \over 4}} \right) \to R\) be defined as $$f(x) = \left\{ {\matrix{
{{{(1 + |\sin x|)}^{{{3a} \over {|\sin x|}}}}} & , & { - {\pi \over 4} < x < 0} \cr
b & , & {x = 0} \cr
{{e^{\cot 4x/\c...

Let \(f:[0,\infty ) \to [0,3]\) be a function defined by \(f(x) = \left\{ {\matrix{ {\max \{ \sin t:0 \le t \le x\} ,} & {0 \le x \le \pi } \cr {2 + \cos x,} & {x > \pi } \cr } } \right.\)Then which of the following is true?

The value of \(\mathop {\lim }\limits_{x \to 0} \left( {{x \over {\root 8 \of {1 - \sin x} - \root 8 \of {1 + \sin x} }}} \right)\) is equal to :

If \(\alpha\), \(\beta\) are the distinct roots of x2 + bx + c = 0, then \(\mathop {\lim }\limits_{x \to \beta } {{{e^{2({x^2} + bx + c)}} - 1 - 2({x^2} + bx + c)} \over {{{(x - \beta )}^2}}}\) is equal to :

If \(\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} - x + 1} - ax} \right) = b\), then the ordered pair (a, b) is :

The value of \(\mathop {\lim }\limits_{h \to 0} 2\left\{ {{{\sqrt 3 \sin \left( {{\pi \over 6} + h} \right) - \cos \left( {{\pi \over 6} + h} \right)} \over {\sqrt 3 h\left( {\sqrt 3 \cosh - \sinh } \right)}}} \right\}\) is :

Let \(f(x) = {\sin ^{ - 1}}x\) and \(g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}\). If \(g(2) = \mathop {\lim }\limits_{x \to 2} g(x)\), then the domain of the function fog is :

Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then \(\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}\) equals :

Let f : R \(\to\) R be defined as \(f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.\) If f(...

Let a, b \(\in\) R, b \(\in\) 0, Define a function \(f(x) = \left\{ {\matrix{ {a\sin {\pi \over 2}(x - 1),} & {for\,x \le 0} \cr {{{\tan 2x - \sin 2x} \over {b{x^3}}},} & {for\,x > 0} \cr } } \right.\). If f is continuous at x ...

\(\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right)\) is equal to :

Let [t] denote the greatest integer less than or equal to t. Let f(x) = x \(-\) [x], g(x) = 1 \(-\) x + [x], and h(x) = min{f(x), g(x)}, x \(\in\) [\(-\)2, 2]. Then h is :

Let f : R \(\to\) R be defined as$$f(x) = \left\{ {\matrix{
{{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5x - {x^2} - 6)}},} & {x < 2} \cr
{{e^{{{\tan (x - 2)} \over {x - [x]}}}},} & {x > 2} \cr
{\mu ,} & {x = 2} \c...

Consider the functionwhere P(x) is a polynomial such that P'' (x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equal to _____________.

\(\mathop {\lim }\limits_{n \to \infty } {\left( {1 + {{1 + {1 \over 2} + ........ + {1 \over n}} \over {{n^2}}}} \right)^n}\) is equal to :