Continuity and Differentiability

$lim_{x\rightarrow 1}\,\left[ \dfrac{x^{2}+1}{x+100} \right]$

If $f(x)=\left\{\begin{array}{ll}x+\lambda, & x<3 \\ 4, & x=3 \\ 3x-5, & x>3\end{array} \right)$ is continuous at $x=3$, then $\lambda=$

$lim_{x\rightarrow\infty}\,\left( \dfrac{x^{3}+3x+2}{(x+4)(x-6)(x-3)} \right)=$

If $f\left( x \right)=\left\{\begin{array}{ll}x-1, & x<0 \\ \dfrac{1}{4}, & x=0,\,then\, \\ x^{2}, & x>0\end{array} \right)$

If the function $f(x)$ defined by $f(x)=\left\{\begin{array}{lll}a x+1 & \text { if } & x \leq 3 \\ b x+3 & \text { if } & x > 3\end{array}\right.$ is continuous at $x=3$, then $(a-b)=\cdots \cdots \cdots \cdot$

Find the value of $a-b$ so that the function defined by $f(x)=\left\{\begin{array}{ll}ax+1, & \,if\,x\leq 3 \\ bx+3, & \,if\,x>3\end{array} \right)$ is continuous at $x=3$.

The number of point / points where the function $f(x)=\dfrac{1}{x^{2}-5|x|+6}$ is discontinuous is......

Evaluate: $lim_{x\rightarrow 3}\,\dfrac{x^{4}-81}{2x^{2}-5x-3}$

Evaluate : $lim_{x\rightarrow\dfrac{\pi}{2}}\,\dfrac{1+\cos 2x}{(\pi-2x)^{2}}$

$lim_{x\rightarrow 0}\,\dfrac{(1-\cos 2x)(3+\cos x)}{x\tan 4x}$ is equal to

$lim_{x\rightarrow 0}\,\dfrac{1-\cos mx}{1-\cos nx}=$

If $f\left( x \right)=\left\{\begin{array}{r}\left( \dfrac{x^{2}}{a} \right)-a,\,when\,x<a \\ 0,\,when\,x=a,\,then\, \\ a-\left( \dfrac{x^{2}}{a} \right),\,when\,x>a\end{array} \right)$

The value of $\lim _{n \rightarrow \infty} \dfrac{(n+2)!+(n+1)!}{(n+2)!-(n+1)!}$ is $\_\_\_\_$

If $f: R \rightarrow R$ is an even function then

The function $\mathrm{f}(x)=2 x-\left|x-x^{2}\right|$ is