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.

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

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.

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.