# How do you solve limits with f notations in them?

1. Dec 6, 2016

### Arnoldjavs3

1. The problem statement, all variables and given/known data
Problem 1:
$f'(1) = -2$
Solve:
$$\lim_{x\to0} \frac{f(e^{5x} - x^2) - f(1)}{x}$$

2. Relevant equations

3. The attempt at a solution
Okay so these type of problems really get to me. I'm going to assume some level of substitution are needed but I'm really unsure.

I'm guessing that I can do something like $u = e^5x - x^2$(or maybe i have to do something like$h(x) = f(e^{5x} - x^2)$.

But what would I do from there? Would f(1) be equal to h(1)? Do I just use l'hopitals if this is on the right path? Is there a standard procedure to follow when dealing with limits that have function notations in them?

Last edited: Dec 6, 2016
2. Dec 6, 2016

### Math_QED

Is this all the information you got? How would you apply Hopital when you don't know that the numerator goes to zero?

3. Dec 6, 2016

### PeroK

Is that supposed to be $e^{5x}$?

4. Dec 6, 2016

### Staff: Mentor

Are you sure it isn't $f(e^{5x} - x^2)$?

5. Dec 6, 2016

### vela

Staff Emeritus
That limit looks to me suspiciously similar to a derivative. If you use $u = e^{5x} - x^2$, you'd have f(u)-f(1) in the numerator. What would you need in the denominator to get the derivative f'(1)?

6. Dec 6, 2016

### Arnoldjavs3

Yeah this is all the information I have. I forgot that L'hopitals can only be used if indeterminate form is found.
Yeah you're right. Sorry about that i'll edit it
Are you saying that x should be equal to f'(1)?

7. Dec 6, 2016

### vela

Staff Emeritus
No. I'm not sure how you jumped to that conclusion.

8. Dec 6, 2016

### Stephen Tashi

You should be able to show that the limit of the numerator is zero and this will justify using l'hospitals rule. . ( i.e. show $lim_{x \rightarrow 0} {f(e^{5x} - x^2) - f(1)} = f(1)-f(1) = 0$ ) To show that, you need to use the fact that if f'(x) exists at x = a then f is continuous at x = a and also you need a result that tells about the limit of a composition of two functions. There is probably a theorem in your text materials that tells some conditions which imply that $lim_{x \rightarrow a} f(g(x)) = f(g(a))$.