Limits and Derivatives

Suppose f(x) = {a+bx, x<1,4,x=1,b-a,x, x>1} and if lim_x→1 f(x) = f(1), what are possible values of a and b?

Evaluate:

lim_x→1 [^{(x - 2)}/_{(x² - x)} - ¹/_{(x³ - 3x² + 2x)}]

If(x) = {mx² + n,   x < 0

              mx + m,   0 ≤ x ≤ 1

              nx² + m,   x > 1

For what integers m and n does both lim_x→0 f(x) and lim_x→1 f(x) exist?

Evaluate: lim_h→0 [((a + h)² sin(a + h) - a² sin a ) / h]

find lim_{x → 1} f(x), where f(x) = {x² - 1, x ≤ 1 -x² - 1, x > 1 

Evaluate: lim_x→2 [(¹/_{x - 2}) - (^{2(2x - 3)}/_{x³ - 3x² + 2x})]

Evaluate: lim_x→0 [(sin(2 + x) - sin(2 - x)) / x]

Evaluate: lim_x→0 [(tan x - sin x) / (sin³ x)]

Evaluate: lim_x→π/2 [(1 - sin x) / cos x]

Find the derivative of f(x) = (2x + 3)/(x - 2) from the first principle.

Evaluate:lim_x→1 ^{(x15 - 1)}/_{(x¹⁰ - 1)}

Evaluate: lim_x→π/2 (secx - tanx)

Evaluate: lim_x→π/6 [(2sin² x + sinx - 1) / (2sin² x - 3sin x + 1)]

Evaluate: lim_{x → 0} (^sin 4x/_sin 2x)