limit of x*tan(1/x) as x goes to infinity, L'Hospital's Rule  Summary and Q&A
TL;DR
Calculating the limit as x approaches infinity of x * tangent(1 / x) using Lobito's rule.
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.