How do i calculate $\lim_{h\to0}\frac{f(a+h^2)-f(a+h)}{h}$?

Question

All do i know about this problem is that f can be derived in "a".

What troubles me is the h squared,i just can't get rid of it or make it useful,no matter what i do, i always end up with it giving me an undefined limit, so it stays like that,any idea on how to get rid of it? or any rule i can use to make this easy?

Answer 1

This is $$\lim_{h\to0}\frac{f(a+h^2)-f(a)}{h}-\lim_{h\to0}\frac{f(a+h)-f(a)}{h}.$$ If $f'(a)$ exists, this is $$\lim_{h\to0}h\left(\frac{f(a+h^2)-f(a)}{h^2}\right)-f'(a)=-f'(a).$$

Answer 2

Hint:$$\lim_{h\to0}\frac{f(a+h^2)-f(a)}h=\lim_{h\to0}h\frac{f(a+h^2)-f(a)}{h^2}=h\times f'(a)=0.$$

Answer 3

You can write $$\frac{f(a + h^2) - f(a+h)}{h} = h \cdot \frac{f(a+h^2) - f(a)}{h^2} - \frac{f(a+h) - f(a)}{h}$$ and compute these two limits individually.