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How to Determine Limits of Sequences with L'Hôpital's Rule

You can use L’Hôpital’s rule to find limits of sequences. L'Hôpital's rule is a great shortcut for when you do limit problems. Here it is:

image0.png

Convergence and Divergence: You say that a sequence converges if its limit exists, that is, if the limit of its terms equals a finite number. Otherwise, the sequence is said to diverge.

What do you think? After going up for a couple terms, the sequence goes down and it appears that it’ll keep going down — looks like it will converge to zero. L’Hôpital’s rule proves it. You use the rule to determine the limit of the function

image1.png

which goes hand in hand with the sequence

image2.png

Keep in mind that to use L’Hôpital’s rule, you take the derivative of the numerator and the derivative of the denominator, and then you replace the numerator and denominator by their respective derivatives.

For this problem, you have to use L’Hôpital’s rule twice:

image3.png

Because the limit of the function is 0, so is the limit of the sequence, and thus the sequence

image4.png

converges to zero.

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