find the value of the constants a and b to ensure the following function is differentiable for all real values of x:

f(x) = { (ax / sqrtx) + b , x less than or equal to 1

and x^2 - 1 , x is greater than 1

What two conditions must be met?

updated sorry didn't know it would look like that

also find a and subsequently the limit of lim (x->-2)[ 3x^2 + ax + a + 3/ x^2 + x - 2] which i found a to be 15 after some trial and error and established that the end result would be lim (x->-2) [(3x + 9)(x+2)/(x-1)(x+2)] = -1 but would like better reasoning for the deduction of a

Originally Posted by fibonacci007
find the value of the constants a and b to ensure the following function is differentiable for all real values of x:

f(x) = { (ax / sqrtx) + b , x less than or equal to 1

and x^2 - 1 , x is greater than 1
Given
$f(x) = \left \{ \begin{matrix} \frac{ax}{\sqrt{x}} + b & x \le 1 \\ x^2 -1 & x > 1 \end{matrix} \right.$

1. Continuity at $x=1$
$\lim_{h \rightarrow 0} \frac{a(1-h)}{\sqrt{(1-h)}} + b = \lim_{h \rightarrow 0} (1+h)^2 -1 \implies a+b =0$

2. Derivative at $x=1$
$\lim_{h \rightarrow 0} \frac{1}{2} \frac{a}{\sqrt{(1-h)}} = \lim_{h \rightarrow 0} 2(1+h) \implies \frac{a}{2} = 2$

$a= 4, b = -4$