Question

How do you find the limit?

Answer 1

You can recognize the limit as a derivative. Recall that

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

and more to the point,

$f'(1)=\displaystyle\lim_{h\to0}\frac{f(1+h)-f(1)}h$

We have

$f(x)=x^4\implies f'(x)=4x^3\implies f'(1)=4$

so the answer is B.

We don't actually have to invoke the definition of the derivative. Instead we can just use the definition of $f(x)$:

$\displaystyle\lim_{h\to0}\frac{(1+h)^4-1^4}h=\lim_{h\to0}\frac{1+4h+6h^2+4h^3+h^4-1}h=\lim_{h\to0}(4+6h+4h^2+h^3)$

which you can see also approaches 4.