# Find Limit using L'Hospital Rule: e^x / (e^x + 7x)

• naspek
In summary, the conversation discusses finding a limit using L'Hospital's rule. The limit is of the form infinity/infinity, so the rule is applied again to get a defined answer of 0. The idea is to take the derivatives of the original function and continue applying the rule until a defined answer is obtained. Checking the answer can also be done by multiplying the expression by exp(-x).
naspek
Find the following limit using L'Hospital rule..

lim...(e^x) / [(e^x) + 7x]
x->infinity

= f'(x) / g'(x)
=e^x / (e^x) + 7
= infinity/infinity
=0
am i got it right?

naspek said:
Find the following limit using L'Hospital rule..

lim...(e^x) / [(e^x) + 7x]
x->infinity

= f'(x) / g'(x)
=e^x / (e^x) + 7
Okay to here

= infinity/infinity
What does that mean?

=0
Why would you think that?

am i got it right?
One you have e^x/(e^x+ 7) either use L'Hospital's rule again or divide both numerator and denominator by e^x.

e^x/e^x = 1 ?

Err... wait. You said: when I take the limit directly I get infinity / infinity, which is not defined. So we need L'Hôpitals rule. I agree with that.
Then you take the derivatives, and get infinity / infinity again. Then suddenly, that is defined and equal to 0?

The idea is, that
$$\lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{f'(x)}{g'(x)}$$
in this case, but on the right hand side you get something which is still of the form "infinity / infinity". So you will need to apply it again.

Afterwards, to check your answer, you could try multiplying numerator and denominator by exp(-x).

## 1. What is the purpose of using L'Hospital's rule in finding limits?

L'Hospital's rule is a mathematical technique used to evaluate limits that are in an indeterminate form, such as 0/0 or ∞/∞. It allows us to simplify the expression and make it easier to evaluate the limit.

## 2. How do you use L'Hospital's rule to find the limit of a function?

To find the limit of a function using L'Hospital's rule, first check if the limit is in an indeterminate form. Then, take the derivative of both the numerator and denominator of the fraction. If the limit is still in an indeterminate form, repeat the process until the limit is no longer indeterminate. The final result will be the limit of the original function.

## 3. Is it always necessary to use L'Hospital's rule to find a limit?

No, L'Hospital's rule is only used when the limit is in an indeterminate form. If the limit can be evaluated by direct substitution or other methods, L'Hospital's rule is not needed.

## 4. Can L'Hospital's rule be used for all types of functions?

No, L'Hospital's rule can only be used for functions that are differentiable. This means that the function must have a well-defined derivative at the point of the limit.

## 5. Are there any limitations or restrictions when using L'Hospital's rule?

One limitation of L'Hospital's rule is that it can only be used to evaluate limits at points where the function is differentiable. Additionally, it should only be used as a last resort when other methods of finding the limit are not possible.

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