Evaluate the following limit \(\lim_ { x\rightarrow 0 }{ lim } \frac { tanx-sinx }{ x } \)

Evaluate the following limit \(\lim_ { x\rightarrow 0 }{ lim } \frac { tan2x-x }{ 3x-sinx } \)

Evaluate the following limit \(\lim_ { x\rightarrow 0 }{ lim } \frac { { x }^{ 2 }cosx }{ 1-cosx } \)

Evaluate the following limits : \(\lim_{ x\rightarrow 1 }{ lim } \frac { { x }^{ m }-1 }{ { x }^{ n }-1 } \)

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) X g(x)

Evaluate the limits \(\lim _{ x\rightarrow 3 }{ { (4x }^{ 3 }-{ 2x }^{ 2 } } -x+1)\)

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

If \(y=\sqrt { \frac { 1-cos2x }{ 1+cos2x } } ,\ x\in (0,\frac { \pi }{ 2 } )\cup (\frac { \pi }{ 2 } ,\pi )\) , then find \(\frac { dy }{ dx } \).

Evaluate \(\lim _{x \rightarrow 0} \frac{e^{\sin x-1}}{x}\)

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

Evaluate \(\lim_ { x\rightarrow 0 }{ lim } \frac { cos2x-1 }{ cosx-1 } \)

Evaluate cot x

Find the derivative of \(\frac { { (px }^{ 2 }+qx+r) }{ ax+b } \) (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers)

Find the derivative of \(\frac { \sin (x+a) }{ \cos x } \) (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers)

Find the derivative of  \(\frac { a+b\sin\ x }{ c+d \cos\ x } \) (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers)