How To Find The Limit At Infinity

TL;DR

Learn how to find the limit as x approaches infinity or negative infinity for different types of functions.

Transcript

in this lesson we're going to talk about how to find a limit at infinity so what is the limit as x approaches infinity for the function x squared so if x became very large what would x squared become so let's say if x was a thousand x squared would turn into a larger value it's going to be one million so as x gets very large this turns into infinit... Read More

Key Insights

• ♾️ The limit as x approaches infinity or negative infinity for x squared is positive infinity, while the limit for positive x cubed is positive infinity as x approaches infinity and negative infinity as x approaches negative infinity.
• 🍉 In polynomial functions, you can ignore insignificant terms and find the limit based on the highest degree term.
• ❣️ Rational functions with bottom-heavy ratios, where the degree of the denominator exceeds the numerator, will always have a limit of zero as x approaches infinity.
• 🗂️ Rational functions with the same degree on top and bottom can have their limits found by dividing the coefficients.

Questions & Answers

Q: How do you find the limit as x approaches negative infinity for a polynomial function?

For polynomial functions, you can ignore insignificant terms and find the limit based on the highest degree term. The limit as x approaches negative infinity of 5 + 2x - x cubed can be simplified to the limit of -x cubed, which gives a final answer of positive infinity.

Q: What is the limit as x approaches infinity for a rational function with a bottom-heavy ratio?

A rational function with a bottom-heavy ratio, where the degree of the denominator is higher than the numerator, will always have a limit of zero as x approaches infinity. For example, the limit as x approaches infinity for 1/x is zero.

Q: How do you find the limit as x approaches infinity for a rational function with the same degree on top and bottom?

If a rational function has the same degree on top and bottom, you can divide the coefficients to find the limit. For example, the limit as x approaches infinity for (5x + 2) / (7x - x squared) is 5/7.

Q: How do you find the limit as x approaches infinity for a rational function that is top-heavy?

For top-heavy rational functions, you can remove insignificant terms in the numerator and find the limit based on the highest degree term. The limit as x approaches infinity for 6x squared divided by 3x simplifies to 2 times infinity, which is positive infinity.

Summary & Key Takeaways

• When finding the limit as x approaches infinity for functions like x squared or positive x cubed, the limit is positive infinity. When finding the limit as x approaches negative infinity, x squared gives a positive infinity while positive x cubed gives a negative infinity.

• For polynomial functions, you can ignore insignificant terms and find the limit based on the highest degree term. For example, the limit as x approaches negative infinity of 5 + 2x - x cubed can be simplified to the limit of -x cubed, which gives a final answer of positive infinity.

• Rational functions with bottom-heavy ratios where the degree of the denominator is higher than the numerator will always have a limit of zero as x approaches infinity. For example, the limit as x approaches infinity for 1/x is zero. If the degrees are the same, you can divide the coefficients to find the limit.

• For rational functions that are top-heavy, you can remove insignificant terms in the numerator and find the limit based on the highest degree term. The limit as x approaches infinity for 6x squared divided by 3x simplifies to 2 times infinity, which is positive infinity.