\(\overset{lim}{x\rightarrow 0} \frac{x}{tan x}\) is _______.

\(\overset{lim}{x\rightarrow 0} \frac{sin2x}{x}\) is _______.

\(\overset{lim}{x\rightarrow 0} \frac{x^{n}-a^{n}}{x-a}\) is equal to ______.

\(\overset{lim}{x\rightarrow 1} \frac{sin\pi x}{x-1}\) is equal to _______.

\(\overset{lim}{x\rightarrow 0} \frac{\sqrt{1+x}-1}{x}\) is equal to _____.

Show that \(\lim _{ x\rightarrow 4 }{ \frac { |x-4| }{ x-4 } } \)does not exist.

Evaluate the following limits
\(\lim _{x \rightarrow 1} \frac{x^{3}-1}{x-1}\)

Find the derivative of 99x at x = 100.

Evaluate the following limits : \(\lim_{ x\rightarrow 0 }{ lim } \frac { { \left( 1+x \right) }^{ n }-1 }{ x } \)

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
\({ x }^{ 4 }(5sinx-3cosx)\)

Evaluate \(\lim_{x \rightarrow 2} \frac{x^{2}-5x+6}{x^{2}-4}\)

Evaluate:\(\underset { x\rightarrow 0 }{ Lt } \frac { log(2+x)+log0.5 }{ x } \)

If f is an even function, then prove that \(\overset{lim f(x)}{x\rightarrow 0^{-}}=\overset{lim f(x)}{x\rightarrow 0^{+}}\)

Evaluate \(\lim_{x\rightarrow 0}\frac{x^{3}cot x}{1-cos x}\)

Differentiate \(\sqrt{sin x}\) w.r.t. x by first principle method.

Let us consider two functions \(f(x)={ x }^{ 2 }+4\)and g(x) = x-3 such that f(x) and g(x) exist at x=5. Find the limit of the following fuctions at x=5.
f(x) - g(f)

Evaluate \(\lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}-2}{x^{2}}\) 

Evaluate \(\lim _{x \rightarrow \pi / 6} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}\)

Differentiate the function \(\cos { \left( { x }^{ 2 }+1 \right) } \) by the first principle.

Evaluate:\(\lim_ { x\rightarrow 0 } \frac { { e }^{ x }-1-x-{ x }^{ 2 } }{ { x }^{ 2 } } \)

Suppose \( f(x)=\left\{\begin{array}{ll} a+b x, & x<1 \\ 4, & x=1 \text { and if } \lim _{x \rightarrow 1} f(x)=f(1) \\ h-a x & x>1 \end{array}\right.\)what are possible values of a and b?

Evaluate: \(\overset{Lt}{x\rightarrow e}\frac{log x-1}{x-e}\)