Examples

L’Hopital’s Rule for sequence limits

(a) What is the limit of ? (b) What is the limit of ? (c) What is the limit of ?

Solution (a) Identify indeterminate form . Change from to and apply L’Hopital:

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(b) Identify indeterminate form . Change from to and apply L’Hopital:

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(c) (1) Identify form and rewrite as :

(2) Change from to and apply L’Hopital:

(3) Simplify:

(4) Consider the limit:

Squeeze theorem

Use the squeeze theorem to show that as .

Solution

(1) We will squeeze the given general term above and below a sequence that we must devise:

(2) We need to satisfy and . Let us study .

(3) Now for the trick. Collect factors in the middle bunch:

(4) Each factor in the middle bunch is so the entire middle bunch is . Therefore:

Now we can easily see that as , so we set and we are done.

Monotonicity

Show that converges.

Solution

(1) Observe that for all .

Because , we know .

Therefore

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(2) Change to and show is decreasing.

New formula: considered as a differentiable function.

Take derivative to show decreasing.

Derivative of :

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(3) Simplify:

Denominator is . Numerator is . So and is monotone decreasing.

Therefore is monotone decreasing as .