evaluating a limit with power series, calculus 2 tutorial

TL;DR

Using power series expansion, the limit of (1 - cos(x)) / (1 + x - e^x) as x approaches zero is 1.

Transcript

we are going to use power series to complete this limit we have the limit as X goes to zero 1 - cosine X over 1 + x minus E to X we are not going to use Lal rule we will use the power series expansion for cosine X and e to the X Center as zero because X is approaching to zero this is how it goes we will have equal to the Limit as X goes to zero and... Read More

Key Insights

• ✊ Power series expansion can be used to simplify and evaluate limits.
• 😑 Power series expansions for cosine and exponentials help simplify the expression in this particular limit evaluation.
• 😑 Terms with x^2 can be factored out to cancel out and simplify the expression.

Questions & Answers

Q: How is the limit (1 - cos(x)) / (1 + x - e^x) evaluated using power series expansion?

The power series expansions for cosine and exponentials are used. By canceling out common terms and factoring out x^2, the expression is simplified, and the limit is found to be 1.

Q: What is the power series expansion for cosine x?

The power series expansion for cosine x includes terms alternating between positive and negative powers of x, with the coefficients given by the factorial of the corresponding power.

Q: Why is the power series expansion centered at zero used for cosine and exponentials in this limit evaluation?

The power series expansion is centered at zero because the limit is being evaluated as x approaches zero. This allows the terms of the series to accurately represent the behavior of the functions near zero.

Q: Can more terms be added to the power series expansions to improve accuracy in the limit evaluation?

Yes, more terms can be added to the power series expansions to improve accuracy. However, in this specific example, the first four terms were deemed sufficient for the evaluation.

Summary & Key Takeaways

• The content explains how to evaluate a limit using power series expansion.

• The author uses the example of the limit (1 - cos(x)) / (1 + x - e^x) as x approaches zero.

• The power series expansions for cosine and exponentials are used to simplify the expression.

• After canceling out common terms and factoring out x^2, the limit is found to be 1.