# Help with understanding of L'Hospitals Rule

• shocklightnin
In summary, the limit of (lnx)^2/x as x approaches infinity can be solved using L'Hospital's rule twice, resulting in a limit of 0. This is because both the numerator and denominator approach infinity, but the numerator grows faster than the denominator.
shocklightnin

## Homework Statement

This was a question from our lecture notes, just not sure how the prof arrived at the answer.

lim x->infinity (lnx)^2/x

## Homework Equations

lim x->infinity (lnx)^2/x
lim x->infinity 2lnx/x

## The Attempt at a Solution

so both the numerator and denominator are going towards infinity, and by L'H it the lim x->infinity 2lnx/x
so this means that the numerator is 'growing' faster than the denominator, a constant x? also, how does one arrive at the conclusion that the limit is 0?

(lnx)^2/x = (2/x)(ln x)

derivative of ln x = 1/x

Go from there. I'm drunk.

shocklightnin said:

## Homework Statement

This was a question from our lecture notes, just not sure how the prof arrived at the answer.

lim x->infinity (lnx)^2/x

## Homework Equations

lim x->infinity (lnx)^2/x
lim x->infinity 2lnx/x

## The Attempt at a Solution

so both the numerator and denominator are going towards infinity, and by L'H it the lim x->infinity 2lnx/x
so this means that the numerator is 'growing' faster than the denominator, a constant x? also, how does one arrive at the conclusion that the limit is 0?

Just apply L'Hospital's rule again since you are still in an indeterminate inf/inf:
$$\frac{2}{x}$$

It should now be pretty sensible that it approaches 0 as x approaches infinity.

RoshanBBQ, thanks! Completely slipped my mind that sometimes we have to apply L'H more than once.

## What is L'Hospital's Rule?

L'Hospital's Rule is a mathematical technique used to evaluate limits involving indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of the ratio of two functions is indeterminate, then the limit of the ratio of their derivatives will be the same.

## When should L'Hospital's Rule be used?

L'Hospital's Rule should only be used when the limit of the ratio of two functions is indeterminate. This means that both the numerator and denominator of the fraction approach 0 or ∞ as the limit is taken.

## How do you apply L'Hospital's Rule?

To apply L'Hospital's Rule, take the derivative of both the numerator and denominator of the fraction. Then, evaluate the limit of the new ratio of derivatives. If the limit still results in an indeterminate form, repeat the process until a finite value is obtained.

## What are the limitations of L'Hospital's Rule?

L'Hospital's Rule can only be used for limits involving indeterminate forms. It cannot be used for limits that do not result in an indeterminate form, or for limits that involve multiple variables.

## Are there any alternatives to L'Hospital's Rule?

Yes, there are other techniques that can be used to evaluate limits, such as the Squeeze Theorem, Taylor Series, or using algebraic manipulation. It is important to understand these alternatives and choose the most appropriate method for a specific problem.

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