Problem 44 - Calculus of a Single Variable

Question

Finding a Limit In Exercises 43 and \(44,\) find the limit. (Hint: Let \(x=1 / t\) and find the limit as \(t \rightarrow 0^{+} . )\) $$ \lim _{x \rightarrow \infty} x \tan \frac{1}{x} $$

Step-by-Step Solution

Answer

The limit of the function as x approaches infinity is 1

Step 1: Substitution

In this step, we substitute 'x' with '1/t' as the hint suggests. Our limit expression becomes \(\lim_{{t \rightarrow 0^{+}}} \frac{\tan(t)}{t}\)

Step 2: L'Hopital's Rule

The current expression still results in an indeterminate form of 0/0 as 't' tends to 0. To tackle this, we apply L'Hopital’s Rule, which states that the limit of the quotient of two functions as x approaches a particular value is the same as the limit of the quotients of their derivatives. Therefore, we differentiate the numerator and denominator with respect to 't'. The expression now reads as \(\lim_{{t \rightarrow 0^{+}}} \frac{\sec^2(t)}{1}\)

Step 3: Evaluate the Limit

Now that we have a proper expression whose limit can be easily determined, we substitute 't' with '0' in the expression, which results in the number 1. Thus, the limit of the given function as x approaches infinity is 1

Key Concepts

L'Hopital's Rule

L'Hopital's Rule is a powerful tool in calculus for finding limits. When you encounter an indeterminate form, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), this rule comes to the rescue. The idea is simple: if you can't directly solve the limit, differentiate the numerator and the denominator separately, then take the limit again. If the new expression is solvable, you have your answer. Here's how you apply it:

• Find the derivatives of the numerator and denominator.
• Form a new limit expression with these derivatives.
• Evaluate the limit of this new expression as the variable approaches a specific value.

In our exercise, substituting \( x=\frac{1}{t} \) changes the expression to \( \frac{\tan(t)}{t} \). This gives us a \( \frac{0}{0} \) form, perfect for L'Hopital's Rule. Applying the rule allows us to find the limit without getting stuck in an indeterminate loop.

Indeterminate Forms

Indeterminate forms occur when plugging in values into a function leads to expressions like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), or even \( \infty - \infty \). These forms aren't directly solvable, which is why they're called 'indeterminate'. They need special methods to evaluate, such as L'Hopital's Rule.

Understanding why these forms are indeterminate is key. For instance, \( \frac{0}{0} \) could potentially equal any number because zero divided by zero is undefined. To solve such problems, we must transform the expression using algebraic tricks or calculus techniques.

Recognizing indeterminate forms quickly helps you decide which method to use when solving limits. It highlights opportunities to simplify the problem and find a meaningful answer.

Trigonometric Limits

Trigonometric limits often appear in calculus, especially when functions involve sine, cosine, or tangent. These limits can be tricky due to oscillations of trigonometric functions as they approach certain angles.

For example, the limit \( \lim_{t \to 0} \frac{\sin(t)}{t} = 1 \) is a fundamental trigonometric limit and is crucial for solving many problems. Its counterpart, \( \lim_{t \to 0} \frac{\tan(t)}{t} \), also equals 1 and is exactly what's used in our problem.

Such special limits help in understanding how trigonometric functions behave near critical points. Keep these standard limits handy as they often simplify seemingly complex problems, revealing straightforward solutions.