limit of x*tan(1/x) as x goes to infinity, L'Hospital's Rule | Summary and Q&A

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February 25, 2015
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limit of x*tan(1/x) as x goes to infinity, L'Hospital's Rule

TL;DR

Calculating the limit as x approaches infinity of x * tangent(1 / x) using Lobito's rule.

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Key Insights

  • ☺️ The limit of x * tangent(1/x) as x approaches infinity initially appears as an indeterminate form of infinity times zero.
  • ♾️ Lobito's rule enables the evaluation of limits in the form of 0/0 or infinity/infinity.
  • 😑 By rewriting the expression, we can transform it into the desired form and use Lobito's rule to simplify and find the limit.

Transcript

how can we handle this question we have the limit as X goes to infinity x * tangent 1 /x so as a good habit let's pluging Infinity into this X and that X first for the first part we have just infinity times we have tangent of 1 over infinity and then one over infinity zero in another word we're talking about tangent of zero and then tangent of Zer ... Read More

Questions & Answers

Q: How do we handle the limit as x approaches infinity of x * tangent(1 / x)?

First, we plug in infinity for x and find that the expression becomes infinity times tangent of zero. Since tangent of zero is zero, we are left with infinity times zero, which is an indeterminate form.

Q: Can we use Lobito's rule immediately to evaluate the limit?

No, Lobito's rule only works for the forms 0/0 or infinity/infinity. In this case, we have infinity times zero, so we need to make some manipulations to transform the expression into one of those forms.

Q: How can we rewrite the expression to employ Lobito's rule?

By considering X as 1/(1/x), we can transform the original expression to the limit as x approaches infinity of (1/x) * tangent(1/x). This allows us to use Lobito's rule since the new form is 0/0.

Q: What does Lobito's rule entail?

Lobito's rule involves differentiating the numerator and denominator separately. The derivative of tangent is secant squared, and the derivative of 1/x is -1/x^2. After simplifying the expression, we can cancel out common terms and evaluate the results.

Summary & Key Takeaways

  • When evaluating the limit as x approaches infinity of x * tangent(1/x), we encounter the indeterminate form of infinity times zero.

  • To handle this, we can rewrite X as 1/(1/x) and apply Lobito's rule, which requires the form to be 0/0 or infinity/infinity.

  • Applying Lobito's rule involves differentiating the numerator and denominator separately and simplifying the expression, eventually resulting in a value of 1 for the limit as x approaches infinity.

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