How to Find Limits at Infinity with Square Roots

TL;DR

To determine the limit as x approaches negative infinity for a rational expression involving square roots, divide the numerator and denominator by the highest degree term. This simplifies the expression to yield a limit of 1, demonstrating the usefulness of this method in calculus.

Transcript

• [Voiceover] Let's see if we can find the limit as x approaches negative infinity of the square root of 4x to the fourth minus x over 2x squared plus three. And like always, pause this video and see if you can figure it out. Well, whenever we're trying to find limits at either positive or negative infinity of rational expressions like this, it's u... Read More

Key Insights

• 😑 Dividing rational expressions by the highest degree term helps to simplify and evaluate limits at positive or negative infinity.
• 🍉 The highest degree term is determined by looking at the terms in the numerator and denominator, considering any radicals.
• 🌥️ As x approaches negative infinity, dividing by larger and larger negative values causes the fractions to approach zero.
• 😑 The limit as x approaches negative infinity can be found by dividing the numerator and denominator by the highest degree term, simplifying the expression, and evaluating the limit.

Questions & Answers

Q: How do you find the limit as x approaches negative infinity for rational expressions?

To find the limit, divide the numerator and denominator by the highest degree term. In this case, we divide by x^2. Simplify the expression and evaluate the limit.

Q: Why is it useful to divide by the highest degree term when finding limits at positive or negative infinity?

Dividing by the highest degree term helps to eliminate any constants or lower degree terms that do not contribute to the limit. It allows us to focus on the terms that have the greatest impact as x approaches infinity or negative infinity.

Q: What happens to the numerator and denominator as x approaches negative infinity?

As x approaches negative infinity, both the numerator and denominator go to zero. This is because we are dividing by larger and larger negative values, causing the fractions to approach zero.

Q: What is the final result of the limit as x approaches negative infinity for the given expression?

The final result is 1. After simplifying the expression and evaluating the limit, we find that the limit as x approaches negative infinity is equal to 1.

Summary & Key Takeaways

• To find the limit as x approaches negative infinity of rational expressions, divide the numerator and denominator by the highest degree term (in this case, x^2).

• Dividing the expression results in the limit being equal to the square root of 4 divided by 2, which simplifies to 1.

• As x approaches negative infinity, both the numerator and denominator go to zero, resulting in the limit equal to 1.