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

a. find f and a, and then evaluate the derivative.

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`.