Limits, Continuity and Differentiability

For a real number $\alpha$, let $[\alpha]$ denote the greatest integer less than or equal to $\alpha$. For a finite set $S$, let $|S|$ denote the number of elements in the set $S$.
Consider the functions $f:(-3,3) \rightarrow(-\infty, \inft...

Consider the function $f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \to (-\infty, \infty)$ defined by\(f(x) = (|x| + |x-1|) \sin x + \left[ x \sin x \right],\)where $\left[ x \sin x \right]$ is the greatest integer less than or equal to $x \...

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ be an arbitrary function and let $g : \mathbb{R} \to \mathbb{R}$ be the function defined by\(g(x) = x f(x), \quad \text{for all } x \in \mathbb{R}.\)Th...

Let $x_0$ be the real number such that $e^{x_0} + x_0 = 0$. For a given real number $\alpha$, define \(g(x) = \frac{3x e^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)}\) for all real numbers $x$. Then which one of the following statements is T...

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let [ x ] denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and cho...

Let α and β be the real numbers such that
$ \lim\limits_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int\limits_0^x \frac{1}{1-t^2} \, dt + \beta x \cos x \right) = 2. $
Then the value of α + β is ___________.

Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f : \mathbb{R} \to \mathbb{R}$ by$f(x)=\left\{\begin{array}{cc}2-2 x^2-x^2 \sin \frac{1}{x} & \text { if } x \neq 0, \\ 2 & \text { if } x=0 .\end{array}\right.$Then ...

Let $S$ be the set of all $(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}$ such that

$$ \lim\limits_{x \rightarrow \infty} \frac{\sin \left(x^2\right)\left(\log _e x\right)^\alpha \sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta}\left(\l...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by

$$ f(x)=\left\{\begin{array}{cc} x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0, \\ 0, & \text { if } x=0 . \end{array}\right. $$

Then which of the follow...

Let $k \in \mathbb{R}$. If $\lim \limits_{x \rightarrow 0+}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6$, then the value of $k$ is

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions defined by

$$ f(x)=\left\{\begin{array}{ll} x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0, \end{array} \quad \text { and } g(x...

Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4 x]\left(x-\frac{1}{4}\right)^2\left(x-\frac{1}{2}\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements i...

For positive integer $n$, define

\(f(n)=n+\frac{16+5 n-3 n^{2}}{4 n+3 n^{2}}+\frac{32+n-3 n^{2}}{8 n+3 n^{2}}+\frac{48-3 n-3 n^{2}}{12 n+3 n^{2}}+\cdots+\frac{25 n-7 n^{2}}{7 n^{2}} .\)

Then, the value of $$\mathop {\lim }\limits_{n \to...

If
\(\beta=\lim \limits_{x \to 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin ^{2} x},\)

then the value of $6 \beta$ is ___________.

Let \(\alpha\) be a positive real number. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g:(\alpha, \infty) \rightarrow \mathbb{R}\) be the functions defined by

$$
f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=...

Let f : R \(\to\) R be defined by \(f(x) = {{{x^2} - 3x - 6} \over {{x^2} + 2x + 4}}\)Then which of the following statements is (are) TRUE?

Let the functions \(f:( - 1,1) \to R\) and \(g:( - 1,1) \to ( - 1,1)\) be defined by \(f(x) = |2x - 1| + |2x + 1|\) and \(g(x) = x - [x]\), where [x] denotes the greatest integer less than or equal to x. Let \(f\,o\,g:( - 1,1) \to R\) be th...

The value of the limit$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{4\sqrt 2 (\sin 3x + \sin x)} \over {\left( {2\sin 2x\sin {{3x} \over 2} + \cos {{5x} \over 2}} \right) - \left( {\sqrt 2 + \sqrt 2 \cos 2x + \cos {{3x} \over 2}} \righ...

Let f : R \(\to\) R and g : R \(\to\) R be functions satisfying f(x + y) = f(x) + f(y) + f(x)f(y) and f(x) = xg(x) for all x, y\(\in\)R. If \(\mathop {\lim }\limits_{x \to 0} g(x) = 1\), then which of the following statements is/are T...

Let the function f : R \(\to\) R be defined by f(x) = x3 \(-\) x2 + (x \(-\) 1)sin x and let g : R \(\to\) R be an arbitrary function. Let fg : R \(\to\) R be the product function defined by (fg)(x) = f(x)g(x). Then which of the follo...

let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit\(\mathop {\lim }\limits_{x \to {0^ + }} {{{{(1 - x)}^{1/x}} - {e^{ - 1}}} \over {{x^a}}}\)is equal to a non-zero real number, is ....

For \(a \in R,\,|a|\, > 1\), let $$\mathop {\lim }\limits_{n \to \infty } \left( {{{1 + \root 3 \of 2 + ...\root 3 \of n } \over {{n^{7/3}}\left( {{1 \over {{{(an + 1)}^2}}} + {1 \over {{{(an + 2)}^2}}} + ... + {1 \over {{{(an + n)}^2}}}} ...

Let f : R be a function. We say that f has PROPERTY 1 if \(\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {\sqrt {|h|} }}\) exists and is finite, and PROPERTY 2 if \(\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {{h^2}}}\) ex...

Let f : R \(\to\) R be given by$$f(x) = \left\{ {\matrix{
{{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 3x + 1,} & {x < 0;} \cr
{{x^2} - x + 1,} & {0 \le x < 1;} \cr
{{2 \over 3}{x^3} - 4{x^2} + 7x - {8 \over 3},} & {1 \le x < 3;} \...

Let f : (0, \(\pi\)) \(\to\) R be a twice differentiable function such that \(\mathop {\lim }\limits_{t \to x} {{f(x)\sin t - f(t)\sin x} \over {t - x}} = {\sin ^2}x\) for all x\(\in\) (0, \(\pi\)).If $$f\left( {{\pi \over 6}} \right...

Let \({f_1}:R \to R,\,{f_2}:\left( { - {\pi \over 2},{\pi \over 2}} \right) \to R,\,{f_3}:( - 1,{e^{\pi /2}} - 2) \to R\) and \({f_4}:R \to R\) be functions defined by(i) \({f_1}(x) = \sin (\sqrt {1 - {e^{ - {x^2}}}} )\),(ii) $${f_2}(x) =...

Let f : R \(\to\) R and g : R \(\to\) R be two non-constant differentiable functions. If f'(x) = (e(f(x) \(-\) g(x))) g'(x) for all x \(\in\) R and f(1) = g(2) = 1, then which of the following statement(s) is (are) TRUE?

For every twice differentiable function \(f:R \to [ - 2,2]\) with \({(f(0))^2} + {(f'(0))^2} = 85\), which of the following statement(s) is(are) TRUE?

The value of \({({({\log _2}9)^2})^{{1 \over {{{\log }_2}({{\log }_2}9)}}}} \times {(\sqrt 7 )^{{1 \over {{{\log }_4}7}}}}\) is ....................

If f : R \(\to\) R is a twice differentiable function such that f"(x) > 0 for all x\(\in\)R, and \(f\left( {{1 \over 2}} \right) = {1 \over 2}\), f(1) = 1, then

Let \(f(x) = {{1 - x(1 + |1 - x|)} \over {|1 - x|}}\cos \left( {{1 \over {1 - x}}} \right)\)for x \(\ne\) 1. Then

Let f : R \(\to\) (0, 1) be a continuous function. Then, which of the following function(s) has (have) the value zero at some point in the interval (0, 1) ?

Let f : R \(\to\) R be a differentiable function such that f(0) = 0, \(f\left( {{\pi \over 2}} \right) = 3\) and f'(0) = 1.If \(g(x) = \int\limits_x^{\pi /2} {[f'(t)\text{cosec}\,t - \cot t\,\text{cosec}\,t\,f(t)]dt}\)for $$x \in \left(...

Let [x] be the greatest integer less than or equals to x. Then, at which of the following point(s) the function \(f(x) = x\cos (\pi (x + [x]))\) is discontinuous?

Let \(f:\left[ { - {1 \over 2},2} \right] \to R\) and \(g:\left[ { - {1 \over 2},2} \right] \to R\) be function defined by \(f(x) = [{x^2} - 3]\) and \(g(x) = |x|f(x) + |4x - 7|f(x)\), where [y] denotes the greatest integer less than or equ...

Let a, b \(\in\) R and f : R \(\to\) R be defined by \(f(x) = a\cos (|{x^3} - x|) + b|x|\sin (|{x^3} + x|)\). Then f is

Let \(\alpha\), \(\beta\) \(\in\) R be such that \(\mathop {\lim }\limits_{x \to 0} {{{x^2}\sin (\beta x)} \over {\alpha x - \sin x}} = 1\). Then 6(\(\alpha\) + \(\beta\)) equals _________.

Let m and n be two positive integers greater than 1. If
\($\mathop {\lim }\limits_{\alpha \to 0} \left( {{{{e^{\cos \left( {{\alpha ^n}} \right)}} - e} \over {{\alpha ^m}}}} \right) = - \left( {{e \over 2}} \right)\)$
then the value of $$...

Let \(g:R \to R\) be a differentiable function with \(g(0) = 0\), \(g'(0) = 0\) and \(g'(1) \ne 0\). Let
\(f(x) = \left\{ {\matrix{ {{x \over {|x|}}g(x),} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.\)
and \(h(x) = {e^{|x|}}\)...

Let f : R \(\to\) R and g : R \(\to\) R be respectively given by f(x) = | x | + 1 and g(x) = x2 + 1. Define h : R \(\to\) R by $$h(x) = \left\{ {\matrix{
{\max \{ f(x),g(x)\} ,} & {if\,x \le 0.} \cr
{\min \{ f(x),g(x)\} ,} & {if\,x ...

The largest value of the non-negative integer a for which \(\mathop {\lim }\limits_{x \to 1} {\left\{ {{{ - ax + \sin (x - 1) + a} \over {x + \sin (x - 1) - 1}}} \right\}^{{{1 - x} \over {1 - \sqrt x }}}} = {1 \over 4}\) is

Let \(f:(a,b) \to [1,\infty )\) be a continuous function and g : R \(\to\) R be defined as $$g(x) = \left\{ {\matrix{
0 & , & {x < a} \cr
{\int_a^x {f(t)dt} } & , & {a \le x \le b} \cr
{\int_a^b {f(t)dt} } & , & {x > b} \cr

...

\(a \in R\) (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 = ?

For every integer n, let an and bn 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

} ...

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

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

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

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