Question

G evaluate lim h → 0 (1 + h)9 − 1 h . hint: this limit represents the derivative of a function f at a given point

Answer 1

$\displaystyle\lim_{h\to0}\frac{(1+h)^9-1}h$

Write $1=1^9$, and recall that for a differentiable function $f(x)$, the derivative at a point $x=a$ is given by

$f'(a)=\displaystyle\lim_{h\to0}\frac{f(a+h)-f(a)}h$

which would suggest that for this limit, $f(x)=x^9$ and $a=1$. We have $f'(x)=9x^8$, and so the value of the limit would be $9(1)^8=9$.