Check if \(lim_{x\rightarrow-58}f(x)\)exists or not, where \(f(x)=\left\{\begin{array}{cc} \frac{|x+5|}{x+5} & , \text { for } x \neq-5 \\ 0, & \text { for } x=-5 \end{array}\right.\)

\(f(x)= \begin{cases}\sin x, & x<0 \\ 1-\cos x, & 0 \leq x \leq \pi \\ \cos x, & x>\pi\end{cases}\)

Evaluate the following limits :
\(lim_{x\rightarrow5}{\sqrt{x-1}-2\over x-5}\)

Evaluate the following limits :\(lim_{x\rightarrow 0}{\sqrt{1+sin x}-\sqrt{1-sinx}\over tanx}\)

State how continuity is destroyed at x = x_ofor each of the following graphs.

Find the slope of the tangent line to the graph of f(x) = 7x + 5 at any point (x₀, f(x₀)).

Differentiate the following: \(y=\sqrt{1+2 \ tan \ x}\)

Find the second order derivative if x and y are given by
x = a cos t
y = a sin t.

If \(y={sin^{-1}x\over \sqrt{1-x^2}}\) , Show that (1 - x²) y₂ - 3x y₁ - y = 0.

If y = \((cos^{-1}x)^2\) ,prove that \((1-x^2){d^2y\over dx^2}-x{dy\over dx}-2=0.\)  Hence find y₂ when x = 0