Understanding Limits of Sequences

The video tutorial explains how to find the limit of a sequence using a theorem that relates the limit of a sequence to the limit of a function as it approaches infinity. It applies this theorem to a specific sequence, analyzing the behavior of its numerator and denominator. The tutorial concludes that the sequence converges to zero, verified by examining a graph of the sequence.

What is the main idea of the theorem used to find the limit of a sequence?

• It relates the limit of a sequence to the limit of a function.

• It defines a sequence as a sum of its terms.

• It states that all sequences converge to zero.

• It explains how to differentiate sequences.

How do we typically determine the limit of a sequence?

• By graphing the sequence and finding its maximum value.

• By finding the derivative of the sequence.

• By summing all terms of the sequence.

• By equating the sequence to a function and analyzing its limit.

What does the theorem imply about the relationship between sequences and functions?

• Functions always have limits, unlike sequences.

• The limit of a sequence can be found using the limit of a related function.

• Sequences and functions have identical limits.

• Sequences are unrelated to functions.

What happens to the cosine function as n approaches infinity?

• It converges to zero.

• It oscillates between -1 and 1.

• It approaches a specific value.

• It becomes undefined.

As n approaches infinity, what happens to 2 raised to the power of n?

• It approaches zero.

• It oscillates between -1 and 1.

• It remains constant.

• It approaches positive infinity.

What is the significance of the closed interval from -1 to 1 in the context of the sequence?

• It is the interval where the sequence diverges.

• It represents the range of the cosine function.

• It is the domain of the sequence.

• It is the range of the sequence's limit.

What is the overall limit of the sequence as n approaches infinity?

• Negative infinity

• Zero

• Infinity

• One

Why does the sequence converge to zero?

• Because the numerator is always zero.

• Because the denominator increases without bound.

• Because the sequence is undefined.

• Because the numerator and denominator both approach infinity.

What does the graph of the sequence show?

• The sequence remains constant.

• The sequence oscillates indefinitely.

• The sequence converges to zero.

• The sequence diverges.

What is the behavior of the sequence's terms as more are generated?

• They approach zero from both positive and negative sides.

• They increase indefinitely.

• They remain constant.

• They decrease to negative infinity.