Limit problem i don't understand

[Sentience]

Homework Statement

lim x ---> infinity (x/x+1)^99

(lim of x over (x+1) all raised to the 99th power)

Homework Equations

The Attempt at a Solution

The answer in my solutions manual is 1. I'm so lost. Any tips would be appreciated.

 

[Sentience]

I know the limit of the inside is one if you use l'hopital but the 99 exponent is throwing me off

 

[eliya]

I haven't taken analysis yet, but I have taken Calculus I-II, and I think what you would do is this:

Divide the numerator and denominator by x. So at the numerator you now have 1. The denominator is now 1 + 1/x. When x goes to infinity 1/x will go to 0. So then you have (1/1)^99 = 1.

 

[Mark44]

Homework Statement

lim x ---> infinity (x/x+1)^99

Use more parentheses!
What you have inside the parentheses would reasonably be interpreted exactly the same as (x/x) + 1. I doubt that's what the problem showed.

The expression you're taking the limit of should be written as (x/(x + 1))⁹⁹

(lim of x over (x+1) all raised to the 99th power)

Homework Equations

The Attempt at a Solution

The answer in my solutions manual is 1. I'm so lost. Any tips would be appreciated.

Your textbook should have some examples where they take the log of the limit expression, and then take the limit.

 

[Sentience]

thank you eliya, that makes sense

Mark, first off sorry about the parentheses, i need to be specific

Second, i remember an example from class now. take the log, and move the 99 in front. From there, do you also use eliya's method of dividing everything by x? ( the (x/(x+1)) inside the log)

 

[lurflurf]

^99 is a continuous function so the limit can be moved inside.
Your book should have a lemma like
lim f(g(x))=f(lim g(x))
lim (g(x))^99=(lim g(x))^99

 

[Sentience]

^99 is a continuous function so the limit can be moved inside.
Your book should have a lemma like
lim f(g(x))=f(lim g(x))
lim (g(x))^99=(lim g(x))^99

I had forgotten that too! Thank you very much

 

[Sentience]

If a function is continuous and I take a limit "inside" can l'hopital's rule be used inside as well?

 

[Mark44]

If the expression you're taking the limit of is suitable for L'Hopital's Rule, yes.

 

[Mark44]

If a function is continuous and I take a limit "inside" can l'hopital's rule be used inside as well?

That doesn't matter. If you have a limit expression that L'Hopital's can be used on, go for it. You've already switched the limit and ln operations.

 

[icystrike]

Divide top and bottom by $$(\frac{1}{x})^{99}}$$ and take the limit ...