91Ó°ÊÓ

When dealing with limits, especially as a variable approaches infinity or zero, we are seeking to understand the behavior of a function as it gets close to a particular point. In the exercise provided, the goal is to evaluate the limit \[ \lim_{x \to \infty} x \tan \left(\frac{1}{x}\right) \]Here, as \(x\) tends towards infinity, \(\frac{1}{x}\) approaches zero. To simplify this, a substitution is used: let \(y = \frac{1}{x}\), making our limit:\[ \lim_{y \to 0^+} \frac{\tan(y)}{y} \]This approach simplifies the evaluation of the limit because it transforms the behavior of the function into a more familiar form—the limit of \( \frac{\tan(y)}{y} \) as \(y\) approaches zero, which is a classic trigonometric limit.

Indeterminate forms occur in calculus when evaluating limits that initially appear unresolved or undefined. The common types are \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). In our exercise, the expression \( \frac{\tan(y)}{y} \) as \(y \to 0\) forms an indeterminate type of \(\frac{0}{0}\), because both the numerator \(\tan(y)\) and the denominator \(y\) approach zero.To resolve this, we apply l'Hospital's Rule, which is used specifically for these types of indeterminate forms. By differentiating the numerator and the denominator separately:

• The derivative of \(\tan(y)\) is \(\sec^2(y)\)
• The derivative of \(y\) is 1

Hence, our limit becomes:\[ \lim_{y \to 0^+} \frac{\sec^2(y)}{1} = \lim_{y \to 0^+} \sec^2(y) = 1 \]Indeterminate forms require special techniques like l'Hospital's Rule to resolve them and find meaningful results.

Trigonometric limits often involve angles approaching zero and need particular attention due to their behavior near these points. The limit \( \lim_{y \to 0^+} \frac{\tan(y)}{y} \) is a fundamental trigonometric limit.This limit is significant because as \(y\) approaches zero, \(\tan(y)\) can be approximated by \(y\) itself since the two functions are very close in value. Therefore, \( \lim_{y \to 0^+} \frac{\tan(y)}{y} = 1 \) is a well-known result.Such trigonometric limits are crucial in calculus since they frequently simplify otherwise complex expressions and are essential to understanding the nature of trigonometric functions at small angles. Knowing these limits inside and out is a valuable tool when tackling calculus problems.