Question

Use the Squeeze Theorem to find the limit of the sequence{4ⁿ/n!}

Answer 1

To find the limit of the sequence {4ⁿ/n!}, we can use the Squeeze Theorem. By comparing the sequence to two other sequences with known limits, we can determine the limit of the original sequence. In this case, the limit is positive infinity.

Step-by-step explanation:

The Squeeze Theorem is used to find the limit of a sequence by comparing it to two other sequences with known limits. In this case, we need to find the limit of the sequence {4ⁿ/n!}. We can use the fact that n! grows faster than any exponential function and compare our sequence to two other sequences.

1. First, note that 4ⁿ refers to a sequence that grows exponentially.
2. Next, note that n! grows factorially, which is even faster than exponential growth.
3. Hence, we can define two more sequences: a lower bound sequence, 0, and an upper bound sequence, 4ⁿ.
4. Now, we can compare the lower and upper bounds to our original sequence and determine their limits:

• The lower bound is 0, which has a limit of 0.
• The upper bound is 4ⁿ, which has a limit of positive infinity.

Since our original sequence is always between the lower and upper bounds, it must also have a limit of positive infinity.