Limits, Continuity and Differentiability

If \(\alpha=\lim _\limits{x \rightarrow 0^{+}}\left(\frac{\mathrm{e}^{\sqrt{\tan x}}-\mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)\) and \(\beta=\lim _\limits{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}\) are the roots of the qu...

For \(\mathrm{a}, \mathrm{b}>0\), let $$f(x)= \begin{cases}\frac{\tan ((\mathrm{a}+1) x)+\mathrm{b} \tan x}{x}, & x< 0 \\ 3, & x=0 \\ \frac{\sqrt{\mathrm{a} x+\mathrm{b}^2 x^2}-\sqrt{\mathrm{a} x}}{\mathrm{~b} \sqrt{\mathrm{a}} x \sqrt{x}},...

Let \([t]\) denote the greatest integer less than or equal to \(t\). Let \(f:[0, \infty) \rightarrow \mathbf{R}\) be a function defined by \(f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]\). Let \(\mathrm{S}\) be the set of all points in the in...

\(\lim _\limits{n \rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\cdots+\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\cdots \cdots+n^3\right)-\left(1^2+2^2+\cdots \cdots+n^2\right)}\) is equal to :

Let \(f\) be a differentiable function in the interval \((0, \infty)\) such that \(f(1)=1\) and \(\lim _\limits{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1\) for each \(x>0\). Then \(2 f(2)+3 f(3)\) is equal to _________.

If the function \(f(x)=\frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^3}, x \in \mathbf{R}\), is continuous at \(x=0\), then \(f(0)\) is equal to :

Let \(\mathrm{a}>0\) be a root of the equation \(2 x^2+x-2=0\). If \(\lim _\limits{x \rightarrow \frac{1}{a}} \frac{16\left(1-\cos \left(2+x-2 x^2\right)\right)}{(1-a x)^2}=\alpha+\beta \sqrt{17}\), where \(\alpha, \beta \in Z\), then $$\al...

Let ,\(f:[-1,2] \rightarrow \mathbf{R}\) be given by \(f(x)=2 x^2+x+\left[x^2\right]-[x]\), where \([t]\) denotes the greatest integer less than or equal to \(t\). The number of points, where \(f\) is not continuous, is :

If \(\lim _\limits{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{\mathrm{m} \sqrt{5}}{\mathrm{n}(2 \mathrm{n})^{2 / 3}}\), where \(\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\), then $$8 \mathrm...

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a function given by
$$f(x)= \begin{cases}\frac{1-\cos 2 x}{x^2}, & x < 0 \\ \alpha, & x=0, \\ \frac{\beta \sqrt{1-\cos x}}{x}, & x>0\end{cases}$$
where \(\alpha, \beta \in \mathbf{R}\). If $$f...

If the function
$$f(x)= \begin{cases}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{cases}$$
is continuous at \(x=0\), then the value of \(a^2\) is equal to

Let \(g(x)\) be a linear function and $$f(x)=\left\{\begin{array}{cl}g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0\end{array}\right.$$, is continuous at \(x=0\). If \(f^{\prime}(1)=f(-1)\), then the value \(g(3)\) ...

\(\lim _\limits{x \rightarrow 0} \frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}\)

If \(\lim _\limits{x \rightarrow 0} \frac{a x^2 e^x-b \log _e(1+x)+c x e^{-x}}{x^2 \sin x}=1\), then \(16\left(a^2+b^2+c^2\right)\) is equal to ________.

Consider the function \(f:(0, \infty) \rightarrow \mathbb{R}\) defined by \(f(x)=e^{-\left|\log _e x\right|}\). If \(m\) and \(n\) be respectively the number of points at which \(f\) is not continuous and \(f\) is not differentiable, then $...

If the function
$$f(x)= \begin{cases}\frac{1}{|x|}, & |x| \geqslant 2 \\ \mathrm{a} x^2+2 \mathrm{~b}, & |x|<2\end{cases}$$
is differentiable on \(\mathbf{R}\), then \(48(a+b)\) is equal to __________.

Let \(f(x)=\sqrt{\lim _\limits{r \rightarrow x}\left\{\frac{2 r^2\left[(f(r))^2-f(x) f(r)\right]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}\) be differentiable in \((-\infty, 0) \cup(0, \infty)\) and \(f(1)=1\). Then the value of ea, such th...

If $\mathrm{a}=\lim\limits_{x \rightarrow 0} \frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $\mathrm{b}=\lim\limits _{x \rightarrow 0} \frac{\sin ^2 x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $a b^3$ is :

Consider the function.
$$ f(x)=\left\{\begin{array}{cc} \frac{\mathrm{a}\left(7 x-12-x^2\right)}{\mathrm{b}\left|x^2-7 x+12\right|} & , x<3 \\\\ 2^{\frac{\sin (x-3)}{x-[x]}} & , x>3 \\\\ \mathrm{~b} & , x=3, \end{array}\right. $$
where $[x]...

\(\text { If } \lim _\limits{x \rightarrow 0} \frac{3+\alpha \sin x+\beta \cos x+\log _e(1-x)}{3 \tan ^2 x}=\frac{1}{3} \text {, then } 2 \alpha-\beta \text { is equal to : }\)

Consider the function \(f:(0,2) \rightarrow \mathbf{R}\) defined by \(f(x)=\frac{x}{2}+\frac{2}{x}\) and the function \(g(x)\) defined by
$$g(x)=\left\{\begin{array}{ll}
\min \lfloor f(t)\}, & 0<\mathrm{t} \leq x \text { and } 0 < x \leq 1 ...

Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, x \neq 0$. If $\mathrm{L}$ and $\mathrm{R}$ respectively denotes the left hand limit and the right hand limi...

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as :
$$ f(x)= \begin{cases}\frac{a-b \cos 2 x}{x^2} ; & x<0 \\\\ x^2+c x+2 ; & 0 \leq x \leq 1 \\\\ 2 x+1 ; & x>1\end{cases} $$

If $f$ is continuous everywhere in $\mathbf{R}$ and $m$ i...

Let $f(x)=\left\{\begin{array}{l}x-1, x \text { is even, } \\ 2 x, \quad x \text { is odd, }\end{array} x \in \mathbf{N}\right.$.
If for some $\mathrm{a} \in \mathbf{N}, f(f(f(\mathrm{a})))=21$, then $\lim\limits_{x \rightarrow \mathrm{a}^...

Let $f(x)=\left|2 x^2+5\right| x|-3|, x \in \mathbf{R}$. If $\mathrm{m}$ and $\mathrm{n}$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $\mathrm{m}+\mathrm{n}$ is equal to :

\(\lim_\limits{x \rightarrow 0}\left(\left(\frac{\left(1-\cos ^{2}(3 x)\right.}{\cos ^{3}(4 x)}\right)\left(\frac{\sin ^{3}(4 x)}{\left(\log _{e}(2 x+1)\right)^{5}}\right)\right)\) is equal to _____________.

Let \(\mathrm{k}\) and \(\mathrm{m}\) be positive real numbers such that the function $$f(x)=\left\{\begin{array}{cc}3 x^{2}+k \sqrt{x+1}, & 0 < x < 1 \\ m x^{2}+k^{2}, & x \geq 1\end{array}\right.$$ is differentiable for all \(x > 0\). The...

If \(\alpha > \beta > 0\) are the roots of the equation \(a x^{2}+b x+1=0\), and $$\lim_\limits{x \rightarrow \frac{1}{\alpha}}\left(\frac{1-\cos \left(x^{2}+b x+a\right)}{2(1-\alpha x)^{2}}\right)^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\b...

Let \(a \in \mathbb{Z}\) and \([\mathrm{t}]\) be the greatest integer \(\leq \mathrm{t}\). Then the number of points, where the function \(f(x)=[a+13 \sin x], x \in(0, \pi)\) is not differentiable, is __________.

Let \(a_{1}, a_{2}, a_{3}, \ldots, a_{\mathrm{n}}\) be \(\mathrm{n}\) positive consecutive terms of an arithmetic progression. If \(\mathrm{d} > 0\) is its common difference, then
$$\lim_\limits{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left...

\(\lim\limits_{x \rightarrow \infty} \frac{(\sqrt{3 x+1}+\sqrt{3 x-1})^6+(\sqrt{3 x+1}-\sqrt{3 x-1})^6}{\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6} x^3\)

Suppose \(f: \mathbb{R} \rightarrow(0, \infty)\) be a differentiable function such that \(5 f(x+y)=f(x) \cdot f(y), \forall x, y \in \mathbb{R}\). If \(f(3)=320\), then \(\sum_\limits{n=0}^{5} f(n)\) is equal to :

Let $f, g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as
$f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{array}\right.$
$g(x)=\left\{\begin{array}{cc}\frac{\sin (x+1)}{(x+1)}, & x \neq-1 \\ 1, & x=-...

Let \(x=2\) be a root of the equation \(x^2+px+q=0\) and \(f(x) = \left\{ {\matrix{ {{{1 - \cos ({x^2} - 4px + {q^2} + 8q + 16)} \over {{{(x - 2p)}^4}}},} & {x \ne 2p} \cr {0,} & {x = 2p} \cr } } \right.\)
Then $$\mathop {\lim }...

The value of \(\mathop {\lim }\limits_{n \to \infty } {{1 + 2 - 3 + 4 + 5 - 6\, + \,.....\, + \,(3n - 2) + (3n - 1) - 3n} \over {\sqrt {2{n^4} + 4n + 3} - \sqrt {{n^4} + 5n + 4} }}\) is :

If the function $$f(x) = \left\{ {\matrix{
{(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 < x < {\pi \over 2}} \cr
\mu & , & {x = {\pi \over 2}} \cr
e^{{{\cot 6x} \over {{}\cot 4x}}} & , & {{\pi \over 2} < x < \pi } ...

Let \(f(x) = \left\{ {\matrix{ {{x^2}\sin \left( {{1 \over x}} \right)} & {,\,x \ne 0} \cr 0 & {,\,x = 0} \cr } } \right.\)
Then at \(x=0\)

\(\mathop {\lim }\limits_{t \to 0} {\left( {{1^{{1 \over {{{\sin }^2}t}}}} + {2^{{1 \over {{{\sin }^2}t}}}}\, + \,...\, + \,{n^{{1 \over {{{\sin }^2}t}}}}} \right)^{{{\sin }^2}t}}\) is equal to

The set of all values of \(a\) for which \(\mathop {\lim }\limits_{x \to a} ([x - 5] - [2x + 2]) = 0\), where [\(\alpha\)] denotes the greatest integer less than or equal to \(\alpha\) is equal to

Let $[x]$ denote the greatest integer function and $f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2$. Let $m$ be the number of points in $[0,2]$, where $f$ is not continuous and $n$ be the number of points in $(0,2)$, where $f$ is not d...

If \(\lim_\limits{x \rightarrow 0} \frac{e^{a x}-\cos (b x)-\frac{cx e^{-c x}}{2}}{1-\cos (2 x)}=17\), then \(5 a^{2}+b^{2}\) is equal to

Let \([x]\) be the greatest integer \(\leq x\). Then the number of points in the interval \((-2,1)\), where the function \(f(x)=|[x]|+\sqrt{x-[x]}\) is discontinuous, is ___________.

Let \(f(x)=\left[x^{2}-x\right]+|-x+[x]|\), where \(x \in \mathbb{R}\) and \([t]\) denotes the greatest integer less than or equal to \(t\). Then, \(f\) is :

Let \(f\) and \(g\) be two functions defined by
$$f(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\ |x-1|, & x \geq 0\end{array}\right.$$ and $$\mathrm{g}(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\ 1, & x \geq 0\end{array}\right.$$
Then $$(g \ci...

Let \(f:( - 2,2) \to R\) be defined by \(f(x) = \left\{ {\matrix{ {x[x],} & { - 2 < x < 0} \cr {(x - 1)[x],} & {0 \le x \le 2} \cr } } \right.\) where \([x]\) denotes the greatest integer function. If m and n respectively are th...

Suppose \(\mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}}\) exists and is equal to L, where
$$F(x) = \left| {\matrix{
{a + \sin {x \over 2}} & { - b\cos x} & 0 \cr
{ - b\cos x} & 0 & {a + \sin {x \over 2}} \cr
0 & {a + ...

The value of \(\mathop {\lim }\limits_{x \to 1} {{({x^2} - 1){{\sin }^2}(\pi x)} \over {{x^4} - 2{x^3} + 2x - 1}}\) is equal to:

The number of points, where the function \(f: \mathbf{R} \rightarrow \mathbf{R}\),
\(f(x)=|x-1| \cos |x-2| \sin |x-1|+(x-3)\left|x^{2}-5 x+4\right|\), is NOT differentiable, is :

If \(\lim\limits_{x \rightarrow 0} \frac{\alpha \mathrm{e}^{x}+\beta \mathrm{e}^{-x}+\gamma \sin x}{x \sin ^{2} x}=\frac{2}{3}\), where \(\alpha, \beta, \gamma \in \mathbf{R}\), then which of the following is NOT correct?

If \([t]\) denotes the greatest integer \(\leq t\), then the number of points, at which the function \(f(x)=4|2 x+3|+9\left[x+\frac{1}{2}\right]-12[x+20]\) is not differentiable in the open interval \((-20,20)\), is __________.