Limit of n!/n^n as n goes to infinity, squeeze theorem, calculus 2 tutorial

TL;DR

The Squeeze Theorem is used to find the limit of n factorial over n to the nth power, which is proven to be equal to zero.

Transcript

by the squeeze something okay we're going to find the limit as n goes to infinity of n factorial over and to the nth power of course let's do the check plugging infinity into all the ends under top we will have infinity factorial when N is a whole number and we will get infinity on the top and then on the bottom of course we also end up with infini... Read More

Key Insights

• 😑 The Squeeze Theorem is a valuable tool for finding limits of expressions that cannot be easily evaluated using other methods.
• 💭 In this analysis, the Squeeze Theorem is used to find the limit of n factorial over n to the nth power.
• 😘 The lower bound of the expression is 0 and the upper bound is 1 over n.
• 😑 By taking the limit as n approaches infinity, it is determined that the expression equals zero.
• 💭 This result demonstrates the relative sizes of n factorial and n to the nth power as n becomes larger.
• 💗 The factorial function grows much slower than the exponential function.
• ❓ The observations and inequalities derived from the Squeeze Theorem provide a rigorous justification for the final result.

Summary & Key Takeaways

• The Squeeze Theorem is applied to find a lower bound and an upper bound for the expression n factorial over n to the nth power.

• The lower bound is found to be 0, while the upper bound is found to be 1 over n.

• Taking the limit as n approaches infinity, it is determined that the expression n factorial over n to the nth power equals zero.