Limits and Derivatives

If f(x)= `{((x - |x|)/x if  x ≠ 0),(2 if x = 0):}` then show that `lim_(x->1)`f(x) does not exist.

Evaluate: `lim_(x->1) ((2x - 3)(sqrtx - 1))/(2x^2 + x - 3)`

Show that the function f(x) = 2x - |x| is continuous at x = 0

Verify the continuity and differentiability of f(x) = `{(1 - x if  x < 1),((1 - x)(2 - x) if 1 <= x <= 2),(3 - x if x > 2):}` at x = 1 and x = 2.

If f(x) = `{(x^2 - 4x if  x >= 2),(x+2 if x < 2):}`, then f(0) is

`lim_(theta->0) (tan theta)/theta` = 

\[\lim_{x->0} \frac{e^x - 1}{x}\]=

For what value of x, f(x) = `(x+2)/(x-1)`  is not continuous?