Engineering Mat...

Given the following statements about a function \(f:R\rightarrow R\), select the right option:

P: If \(f(x)\) is continuous at \(x=x_0\), then it is also differentiable at \(x=x_0\)

Q: If \(f(x)\) is continuous at \(x=x_0\), then it may not be differentiable at \(x=x_0\)

R: If \(f(x)\) is differentiable at \(x=x_0\), then it is also continuous at \(x=x_0\)

Evaluate \(\mathop {\lim }\limits_{x \to 0} {\left[ {\tan \left( {\frac{\pi }{4} + x} \right)} \right]^{1/x}}\)

Let \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {{x^2},\;\;\;if\;\;x = 2}\\ {{e^{ax}},\;\;\;if\;\;x \ne 2} \end{array}} \right.\) ; for the function to be continuous at x = 2 the value of a is

Evaluate \(\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{{e^x}}}{\pi }} \right)^{1/x}}\)

If function \(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} {3ax + b,\;for\;x > 1}\\ {11,\;for\;x = 1}\\ {5ax - 2b,\;for\;x < 1} \end{array}} \right.\) is continuous at x = 1 then which of the following is true?

\(\mathop {\lim }\limits_{x \to 0} \left[ {\frac{{\sin \left( {{x^m}} \right)}}{{{{\sin }^n}\left( x \right)}}} \right]\), m and n are finite natural numbers. What conditions should be imposed on m and n such that the above limit exists?

The value of integral given below is

\(\mathop \smallint \nolimits_0^\pi {x^2}\cos x\;dx\)