Limits at infinity of quotients (Part 2) | Limits and continuity | AP Calculus AB | Khan Academy

TL;DR

Determine the limit of a function as x approaches infinity or negative infinity by identifying the dominating terms in the numerator and denominator.

Transcript

Let's do a few more examples of finding the limit of functions as x approaches infinity or negative infinity. So here I have this crazy function. 9x to the seventh minus 17x to the sixth, plus 15 square roots of x. All of that over 3x to the seventh plus 1,000x to the fifth, minus log base 2 of x. So what's going to happen as x approaches infinity?... Read More

Key Insights

• ♾️ Identifying the dominating terms in the numerator and denominator is crucial when finding limits as x approaches infinity or negative infinity.
• ✋ If the highest degree term is in the numerator, the limit is infinity.
• ✋ If the highest degree term is in the denominator, the limit is 0.
• 👻 Canceling out the dominating terms simplifies the function and allows for easier evaluation of the limit.
• ⛔ Graphing or substituting numbers can help verify the correctness of the calculated limits.

Questions & Answers

Q: How do we determine the limit of a function as x approaches infinity or negative infinity?

To find the limit, we identify the dominating terms in the numerator and denominator. The dominating term in each determines the behavior of the function as x approaches infinity or negative infinity. By canceling out these dominating terms, we can simplify the function and evaluate the remaining expression for the limit.

Q: What happens if the highest degree term is in the numerator?

If the highest degree term is in the numerator, the function's value will approach infinity as x approaches infinity. This is because the numerator grows faster than the denominator, causing the ratio to become unbounded.

Q: What happens if the highest degree term is in the denominator?

If the highest degree term is in the denominator, the function's value will approach 0 as x approaches infinity or negative infinity. In this case, the denominator grows faster than the numerator, causing the ratio to tend towards 0.

Q: How can we verify the results of finding limits using dominating terms?

We can graph the function or substitute large numbers into the expression to verify that the limit matches our calculations. This helps ensure the correctness of our analysis.

Summary & Key Takeaways

• By identifying the terms that dominate in the numerator and denominator, we can simplify the function and determine its limit as x approaches infinity or negative infinity.

• The dominating term in the numerator is the highest degree term, and in the denominator, it is also the highest degree term.

• The limit as x approaches infinity or negative infinity can be found by canceling out the dominating terms and evaluating the remaining expression.