How do you solve limits with f notations in them?

[Arnoldjavs3]

Homework Statement

Problem 1:
$f'(1) = -2$
Solve:
$$\lim_{x\to0} \frac{f(e^{5x} - x^2) - f(1)}{x}$$

Homework Equations

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'hospital's if this is on the right path? Is there a standard procedure to follow when dealing with limits that have function notations in them?

 

[member 587159]

Homework Statement

Problem 1:
$f'(1) = -2$
Solve:
$$\lim_{x\to0} \frac{f(e^5x - x^2) - f(1)}{x}$$

Homework Equations

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?

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

 

[PeroK]

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

 

[Mark44]

Homework Statement

Problem 1:
$f'(1) = -2$
Solve:
$$\lim_{x\to0} \frac{f(e^5x - x^2) - f(1)}{x}$$

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

Homework Equations

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'hospital's if this is on the right path? Is there a standard procedure to follow when dealing with limits that have function notations in them?

 

[vela]

Homework Statement

Problem 1:
$f'(1) = -2$
Solve:
$$\lim_{x\to0} \frac{f(e^{5x} - x^2) - f(1)}{x}$$

Homework Equations

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'hospital's if this is on the right path? Is there a standard procedure to follow when dealing with limits that have function notations in them?

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)?

 

[Arnoldjavs3]

Yeah this is all the information I have. I forgot that l'hospital's can only be used if indeterminate form is found.

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

Yeah you're right. Sorry about that i'll edit it

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)?

Are you saying that x should be equal to f'(1)?

 

[vela]

Are you saying that x should be equal to f'(1)?

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

 

[Stephen Tashi]

Homework Statement

Problem 1:
$f'(1) = -2$
Solve:
$$\lim_{x\to0} \frac{f(e^{5x} - x^2) - f(1)}{x}$$

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 $lim_{x \rightarrow a} f(g(x)) = f(g(a))$.