Determine whether the limit yields the derivative of a differentiable function f. Explain, use algebra to justify your conclusions.

Question

Determine whether the limit yields the derivative of a differentiable function $f$. Explain, use algebra to justify your conclusions. $h$ approaches $0$ in all limits.

1. $\lim_{h\to 0} \frac{f(x+2h)-f(x)}{2h}$

2. $\lim_{h\to 0} \frac{f(x+2)-f(x)}{2h}$

3. $\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}$

4. $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$

Answer 1

You have to use algebra and variable substitutions to make each limit a limit of the form $$\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}$$ Then it will be the derivative of $f(x)$. If it cannot be made to look like this, then it is not the derivative of $f(x)$. I will do the first one to show the concept, and leave the others as an exercise. We have $f(x+2h)$ and we want to have $f(x+h)$, so we substitute $h'=2h$. As $h\rightarrow0$, $h'=2h\rightarrow0$, so the limit becomes $$\lim_{h'\rightarrow0}\frac{f(x+h')-f(x)}{h'}$$ which is the derivative of $f(x)$.

Answer 2

For the second one, the numerator is constant with respect to h, so what we are looking at there is $$\lim_{h\rightarrow0} \frac{C}{2h}$$ which blows up and does not go to the derivative.

For the third, see comments on this reply. (This post has been edited)

The fourth one is what you gave as a definition of the derivative in a comment, so I'll leave that alone.