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Limits and sequences

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a. Use the definition of limits to show that

b. Compute the following limits (justify your answers).
1. ( )

2.

c. Let ( be a sequence. Show that ( is convergent if and only if the sequence ( and ( are convergent to the same limit.

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Solution Summary

This uses the definition of limits in a proof, shows how to compute limits, and shows how to determine if a sequence is convergent.

Solution Preview

Please see the attachment.

Problem (a)
Proof:
For any , let , then for any , we have

Therefore, .

Problem(b)
1.
Proof: Since , then ...

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