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

• Arnoldjavs3
In summary, the problem involves finding the limit of a composition of two functions, where one function is differentiable and the other is continuous. To solve, we can use l'hospitals rule after showing that the limit of the numerator is zero using the fact that f is continuous at x = a if f'(x) exists at x = a.
Arnoldjavs3

## Homework Statement

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

## 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:
Arnoldjavs3 said:

## Homework Statement

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

## 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?

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

Arnoldjavs3
Arnoldjavs3 said:

## 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)##?
Arnoldjavs3 said:

## 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?

Arnoldjavs3 said:

## Homework Statement

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

## 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?
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)?

Yeah this is all the information I have. I forgot that L'hopitals can only be used if indeterminate form is found.
Mark44 said:
Are you sure it isn't ##f(e^{5x} - x^2)##?
Yeah you're right. Sorry about that i'll edit it
vela said:
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)?

Arnoldjavs3 said:
Are you saying that x should be equal to f'(1)?
No. I'm not sure how you jumped to that conclusion.

Arnoldjavs3 said:

## 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))##.

lurflurf

## 1. What is the definition of a limit in f notation?

The limit of a function f(x) as x approaches a is the value that f(x) approaches as x gets closer and closer to a.

## 2. How do you determine the limit of a function using f notation?

To find the limit of a function using f notation, evaluate f(x) as x approaches the given value, a. If the left and right hand limits are equal, then that value is the limit. If they are not equal, then the limit does not exist.

## 3. What is the difference between a one-sided limit and a two-sided limit in f notation?

A one-sided limit only considers the values of f(x) as x approaches a from one direction (either the left or the right). A two-sided limit considers the values of f(x) as x approaches a from both the left and right directions.

## 4. How do you solve a limit with a removable discontinuity in f notation?

If there is a removable discontinuity in the function, you can remove the discontinuity by simplifying the expression or filling in the hole. Once the discontinuity is removed, you can then use the limit definition to find the limit of the function.

## 5. Can a limit in f notation ever be undefined?

Yes, a limit can be undefined if the function has a vertical asymptote or if the left and right hand limits are not equal. In these cases, the limit does not exist.

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