Limits, Continuity & Differentiability - PYQ Math

If \(f(x)=\left\{\begin{array}{cc}\frac{e^{\alpha x}-e^x-x}{x^2}, & x \neq 0 \\ \frac{3}{2}, & x=0\end{array}\right.\)

Find the value of \(\alpha\) for which the function \(f\) is continuous

If \(f^{\prime \prime}(x)\) is continuous at \(x=0\) and \(f^{\prime \prime}(0)=4\), then find the following value. \(\lim _\limits{x \rightarrow 0} \frac{2 f(x)-3 f(2 x)+f(4 x)}{x^2}\) is equal to

The value of \(k(k > 0)\), for which the function \(f(x)=\frac{\left(e^x-1\right)^4}{\sin \left(\frac{x^2}{k^2}\right) \log \left(1+\frac{x^2}{2}\right)}\), where \(x \neq 0\) and \(f(0)=8\)

$$f(x)=\left\{\begin{array}{cc} \[\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\\] K \log 2 \log 3, & x=0 \end{array}\right.$$

Find the value of \(k\) for which the function \(f\) is continuous.

\(\lim _\limits{z \rightarrow 1} \frac{z^{(1 / 3)}-1}{z^{(1 / 6)}-1}\) is equal to

If the function \(f(x)\), defined below is continuous in the interval \([0, \pi]\), then \(f(x)=\left\{\begin{array}{cc}x+a \sqrt{2}(\sin x) & , \quad 0 \leq x < \frac{\pi}{4} \\ 2 x(\cot x)+b, & \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a(\cos 2 x)-b(\sin x), & \frac{\pi}{2} < x \leq \pi\end{array}\right.\)

If \(\lim _\limits{x \rightarrow 0}\left(\frac{11 x^3-3 x+4}{13 x^3-5 x^2-7}\right)=\frac{a}{b}\), then the value of \(a+b\) equals

If \(f(x)=\frac{\log _e\left(1+x^2(\tan x)\right)}{\sin x^3}, x \neq 0\) is to be continuous at \(x=0\), then \(f(0)\) must be equal to

\(\lim _\limits{x \rightarrow 1} \frac{(1-x)\left(1-x^2\right) \ldots\left(1-x^{2 n}\right)}{\left\{(1-x)\left(1-x^2\right) \ldots \ldots\left(1-x^n\right)\right\}^2}=\) _____________, \(\forall n \in N\)

\(\mathop {\lim }\limits_{n \to \infty } {{n{{(2n + 1)}^2}} \over {(n + 2)({n^2} + 3n - 1)}}\) is equal to

If the function \(f(x)\), defined below, is continuous on the interval \([0,8]\), then \(f(x)=\left\{\begin{array}{cc}x^2+a x+b & , \quad 0 \leq x < 2 \\ 3 x+2, & 2 \leq x \leq 4 \\ 2 a x+5 b & , 4 < x \leq 8\end{array}\right.\)

If \(f(x)\), defined below, is continuous at \(x=4\), then

$$f(x) = \left\{ {\matrix{ \[{{{x - 4} \over {|x - 4|}} + a} & , & {x < 4} \cr\] {a + b} & , & {x = 4} \cr \[{{{x - 4} \over {|x - 4|}} + b} & , & {x > 4} \cr\] } } \right.$$

If \([\cdot]\) denotes greatest integer function, then \(\lim _\limits{x \rightarrow \frac{-3}{5}} \frac{1}{\dot{x}}\left[\frac{-1}{x}\right]=\)

Let \(f(x)=\left\{\begin{array}{cl}\frac{1}{|x|}, & \text { for }|x|>1 \\ a x^2+b, & \text { for }|x| \leq 1\end{array}\right.\). If \(\lim _\limits{x \rightarrow 1^{+}} f(x)\) and \(\lim _\limits{x \rightarrow 1^{-}} f(x)\) exist, then the possible values for \(a\) and \(b\) are

$$\frac{d}{d x}\left(\lim _{x \rightarrow 2} \frac{1}{y-2}\left(\frac{1}{x}-\frac{1}{x+y-2}\right)\right)=$$

If \(f(x)=\left\{\begin{array}{cc}\frac{x^2 \log (\cos x)}{\log (1+x)} & , \quad x \neq 0 \\ 0 & , x=0\end{array}\right.\), then at \(x=0, f(x)\) is

If \(l, m(l< m)\) are roots of \(a x^2+b x+c=0\), then \(\lim _\limits{x \rightarrow \alpha} \frac{\left|a x^2+b x+c\right|}{a x^2+b x+c}=\)

Let \(f: R^{+} \longrightarrow R^{+}\) be a function satisfying \(f(x)-x=\lambda\) (constant), \(\forall x \in R^{+}\) and \(f(x f(y))=f(x y)+x, \forall x, y, \in R^{+}\). Then, \(\lim _\limits{x \rightarrow 0} \frac{(f(x))^{1 / 3}-1}{(f(x))^{1 / 2}-1}=\)

$$\begin{aligned} & \text { If } \lim _{x \rightarrow 0} \frac{|x|}{\sqrt{x^4+4 x^2+5}}=k \\ & \lim _{x \rightarrow 0} x^4 \sin \left(\frac{1}{3 \sqrt{x}}\right)=l \text {. Then, } k+l= \end{aligned}$$

If \(\lim _\limits{n \rightarrow \infty} x^n \log _e x=0\), then \(\log _x 12=\)