Examine the following functions for continuity at indicated points
\(f(x)=\left\{\begin{array}{cl} \frac{x^2-4}{x-2}, & \text { if } x \neq 2 \\ 0, & \text { if } x=2 \end{array} \right.\) at x = 2

If y = acos mx + b sin mx, then show that y₂ + m²y = 0.

If y = sin (log x), then show that x²y₂ + xy₁ + y = 0.

Find the derivative of the following functions from first principle. log (x + 1)

Verify the continuity and differentiability of \(f(x)= \begin{cases}1-x & \text { if } x<1 \\ (1-x)(2-x) & \text { if } 1 \leq x \leq 2 \\ 3-x & \text { if } x>2\end{cases}\) at x = 1 and x = 2