Limits, Continuity and Differentiability · Mathematics · AP EAPCET

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

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

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...

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 $$\alpha$$ f...

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}\righ...

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} \...

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

If $$f(x)$$, defined below, is continuous at $$x=4$$, then $$f(x) = \left\{ {\matrix{ {{{x - 4} \over {|x - 4|}} + a} & , & {x 4} \cr } } \rig...