Help With Calculus Limits : Calculus Explained  Summary and Q&A
TL;DR
Learn two techniques for calculating calculus limits using factoring and L'Hopital's Rule.
Questions & Answers
Q: Why can't we simply plug in 1 for x in the first example?
Plugging in 1 results in a zero in the denominator, causing the expression to be undefined. We need to find another way to evaluate the limit.
Q: How does factoring help in the first example?
By factoring the numerator as (x + 1)(x + 2), we can cancel out the common (x + 1) terms in the numerator and denominator. This simplifies the expression, allowing us to substitute 1 to find the limit.
Q: What is L'Hopital's Rule?
L'Hopital's Rule states that if the numerator and denominator of a limit both approach infinity or zero, the limit can be found by taking the derivative of the numerator and denominator and evaluating the new expression.
Q: How does L'Hopital's Rule help in the second example?
The second example involves the limit of x^2 divided by e^x as x approaches positive infinity. Applying L'Hopital's Rule, we take the derivative of the numerator (2x) and denominator (e^x), simplifying the limit to 2/∞, which equals 0.
Summary & Key Takeaways

In calculus limits, plugging in a value may not always work, requiring the use of tricks like factoring.

Factoring can help simplify the expression and cancel out common terms in the numerator and denominator.

Another technique, L'Hopital's Rule, allows you to take the derivative of the numerator and denominator when both approach infinity to simplify the limit.