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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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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