Limits Continuity And Differentiability - Calculus - Mathematics Previous Year Questions

Define $ f: R \rightarrow R $ by $ f(x)=\left\{\begin{array}{cl}\frac{1-\cos 4 x}{x^{2}}, & x < 0 \\ a, & x=0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x > 0\end{array}\right. $

Then, the value of $ a $ so that $ f $ is continuous at $ x=0 $ is

If $f(x)=\frac{x\left(a^x-1\right)}{1-\cos x}$ and $g(x)=\frac{x\left(1-a^x\right)}{a^x\left(\sqrt{1-x^2}-\sqrt{1+x^2}\right)}$, then $\lim _{x \rightarrow 0}(f(x)-g(x))=$

If $f(x)=\left\{\begin{array}{cc}\frac{a \sin x-b x+c x^2+x^3}{2 \log (1+x)-2 x^3+x^4} & , x \neq 0 \\ 0 & , x=0\end{array}\right.$

is continuous at $x=0$, then

If the function $g(x)=\left\{\begin{array}{cl}K \sqrt{x+1} & , 0 \leq x \leq 3 \\ m x+2 & , 3 < x \leq 5\end{array}\right.$ is differentiable, then $K+m=$

If $[x]$ is the greatest integer function, then

$$\mathop {\lim }\limits_{x \to 3} \frac{(3-|x|+\sin |3-x|) \cos [9-3 x]}{|3-x|[3 x-9]}$$

Let ' $a$ ' be a positive real number. If a real valued function

$f(x)=\left\{\begin{array}{cl}\frac{6^x-3^x-2^x+1}{1-\cos \left(\frac{x}{a}\right)} & \text { if } x \neq 0 \\ \log 3 \log 4 & \text { if } x=0\end{array}\right.$ is continuous at $x=0$, then $a=$

If $[t]$ represents the greatest integer $\leq t$, then the value of $\lim\limits_{x \rightarrow 3} \frac{11-[2-x]}{[x+10]}$ is

If the real valued function

$$f(x)=\left\{\begin{array}{ccc} \frac{\cos 3 x-\cos x}{x \sin x}, & \text { if } & x<0 \\ p, & \text { if } & x=0 \\ \frac{\log (1+q \sin x)}{x}, & \text { if } & x>0 \end{array}\right.$$

is continuous at $x=0$, then $p+q=$

The value of $x$ at which the real valued function $f(x)=7|2 x+1|-19|3 x-5|$ is not differentiable is

For $a \neq 0$ and $b \neq 0$, if the real valued function $f(x)=\frac{\sqrt[5]{a(625+x)}-5}{\sqrt[4]{625+b x}-5}$ is continuous at $x=0$, then $f(0)=$