Help With Calculus Limits : Calculus Explained | Summary and Q&A

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November 8, 2012
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Help With Calculus Limits : Calculus Explained

TL;DR

Learn two techniques for calculating calculus limits using factoring and L'Hopital's Rule.

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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.

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