How to Evaluate a Limit Involving Infinity using L'Hopital's Rule

In summary, the conversation discusses the calculation of a limit involving the expression \lim_{x->\pm\infty} xe^{\frac{2}{x}}-x. The individuals involved use algebraic manipulations and the concept of limits to determine the correct answer, which is 2.
  • #1
Petrus
702
0
Hello MHB,
\(\displaystyle \lim_{x->\pm\infty}xe^{\frac{2}{x}}-x\)
I start to divide by x and we know that \(\displaystyle \lim_{x->\pm\infty} \frac{2}{x}=0\)
with other words we get \(\displaystyle 1-1=0\) but that is wrong, how do I do this :confused:

Regards,
\(\displaystyle |\pi\rangle\)
 
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  • #2
Re: limit

Rewrite: \(\displaystyle \dfrac{e^{2/x}-1}{1/x}\)
 
  • #3
Re: limit

tkhunny said:
Rewrite: \(\displaystyle \dfrac{e^{2/x}-1}{1/x}\)
Thanks solved it now!:) got 2 as answer now!:)

Regards,
\(\displaystyle |\pi\rangle\)
 
  • #4
Re: limit

\(\displaystyle x(e^{\frac{2}{x}}-1) = x ( 1+\frac{2}{x}+\frac{2}{x^2}+\mathcal{o}(\frac{1}{x^2})-1 )\)

\(\displaystyle x( \frac{2}{x}+\frac{2}{x^2}+\mathcal{o} (\frac{1}{x^2}) ) = 2+\frac{2}{x}+\mathcal{o} (\frac{1}{x}) \)
 

FAQ: How to Evaluate a Limit Involving Infinity using L'Hopital's Rule

What is a limit involving infinity?

A limit involving infinity refers to the behavior of a mathematical function as the input value approaches infinity. It is used to describe the behavior of a function as it reaches extremely large or small values.

How do you find the limit involving infinity?

To find the limit involving infinity, you can use several methods such as the limit laws, L'Hopital's rule, or substitution. It is important to determine the type of infinity (positive or negative) and the degree of the polynomial in the function to determine the appropriate method.

What does it mean if the limit involving infinity does not exist?

If the limit involving infinity does not exist, it means that the function does not approach a specific value as the input value approaches infinity. This could occur if the function oscillates between different values or if it approaches different values from different directions.

How do limits involving infinity relate to asymptotes?

Limits involving infinity are closely related to asymptotes. An asymptote is a line that a function approaches but never touches. In the case of limits involving infinity, the asymptote is the value that the function approaches as the input value gets larger and larger.

Can a limit involving infinity be infinite?

Yes, a limit involving infinity can be infinite. This occurs when the function approaches a positive or negative infinity as the input value approaches infinity. It is important to note that this does not mean the limit is undefined, but rather that it has a specific value of infinity.

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