Infinite limits

[andres9588]

Have a final project due and dont quite understand this question, please help!

 

[DrPhil]

Have a final project due and dont quite understand this question, please help! View attachment 2807

Finding the limit as x --> infinity means finding the asymptotic behavior of the function at very large (positive) values of x. There are at least two ways to do this, depending on what theorems you have been taught. Can you show us what you have thought of?

 

[andres9588]

Finding the limit as x --> infinity means finding the asymptotic behavior of the function at very large (positive) values of x. There are at least two ways to do this, depending on what theorems you have been taught. Can you show us what you have thought of?

So far ive only came up with solving whats in the brackets, im unsure of what to do next :???: btw, i usually solve limits using the L' Hospital tactic

 

[JeffM]

So far ive only came up with solving whats in the brackets, im unsure of what to do next :???: btw, i usually solve limits using the L' Hospital tactic

Do you know this theorem:

\(\displaystyle \displaystyle \lim_{x \rightarrow \infty}f(x) = a\ and\ \lim_{x \rightarrow \infty}g(x) = b \implies \lim_{x \rightarrow \infty}\{f(x) + g(x)\} = a + b?\)

Using L'Hospital's Rule, what is \(\displaystyle \displaystyle \lim_{x \rightarrow \infty}\dfrac{2 - 3x}{x^2 - 5x + 3}?\)

So how do you attack \(\displaystyle \displaystyle \lim_{x \rightarrow \infty}\dfrac{2x^2 - 1}{(3x + 2)(5x - 3)}?\) At least two ways.

 

[andres9588]

Do you know this theorem:

\(\displaystyle \displaystyle \lim_{x \rightarrow \infty}f(x) = a\ and\ \lim_{x \rightarrow \infty}g(x) = b \implies \lim_{x \rightarrow \infty}\{f(x) + g(x)\} = a + b?\)

Using L'Hospital's Rule, what is \(\displaystyle \displaystyle \lim_{x \rightarrow \infty}\dfrac{2 - 3x}{x^2 - 5x + 3}?\)

So how do you attack \(\displaystyle \displaystyle \lim_{x \rightarrow \infty}\dfrac{2x^2 - 1}{(3x + 2)(5x - 3)}?\) At least two ways.

so you suggest i do them one at a time?

 

[HallsofIvy]

Yes, I believe that is what he is saying!

(And I must say that I dislike the way the problem is originally written. You want to say "\(\displaystyle \lim f(x)= \)", NOT "\(\displaystyle \lim = f(x)\)"!

 

[DrPhil]

So far ive only came up with solving whats in the brackets, im unsure of what to do next :???: btw, i usually solve limits using the L' Hospital tactic

The two terms in the brackets should be solved separately .. one will produce a number and the other will be 0.

l'Hospital can be used because the rational expressions are indefinite, infinity/infinity. However it is not the easiest method. When dealing with polynomials there is another rule you can use.

 

[JeffM]

so you suggest i do them one at a time?

That is not the only way, but it is a straightforward way: break down a complex problem into several simpler ones. In this case, break it down into two simpler problems.

As Dr. Phil has indicated, L'Hospital's rule will work for the simpler problems (with a slight twist for one), but there is probably a simpler way to go for each.