Problem 261 - Calculus Volume 1

Question

Evaluate the limit. \(\lim _{x \rightarrow \infty} \frac{1}{3 x+6}\)

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Type of Limit

We need to evaluate the limit \( \lim_{x \to \infty} \frac{1}{3x + 6} \). This is a limit as \( x \) approaches infinity, indicating we should examine the behavior of the expression as the values of \( x \) grow very large.

Step 2: Simplify the Expression

Given the expression \( \frac{1}{3x + 6} \), we can simplify to better understand the behavior for large \( x \) by dividing the numerator and denominator by \( x \). This gives us \( \frac{1/x}{3 + 6/x} \).

Step 3: Evaluate the Limit of Each Term

As \( x \to \infty \), \( \frac{1}{x} \to 0 \) and \( \frac{6}{x} \to 0 \). Thus, the expression \( \frac{1/x}{3 + 6/x} \) simplifies to \( \frac{0}{3 + 0} \).

Step 4: Calculate the Final Limit

The expression \( \frac{0}{3 + 0} \) results in \( 0 \). Therefore, the original limit \( \lim_{x \to \infty} \frac{1}{3x + 6} \) equals 0.

Key Concepts

Infinite Limits

Infinite limits occur when we talk about the behavior of a function as the variable approaches infinity or negative infinity. These limits help us understand the long-term behavior of functions. When working with these types of limits, it's important to recognize that as the variable gets very large, certain parts of the function might have negligible effects.

• If a variable like \( x \) approaches infinity, terms like \( \frac{1}{x} \) tend to zero, since dividing one by a very large number gives an extremely small number.
• This means that when simplifying expressions involving large \( x \), you can often ignore smaller terms in the denominator unless they significantly affect the outcome.

In our example, as \( x \to \infty \), we're focusing on how the expression \( \frac{1}{3x + 6} \) behaves. The infinite limit tells us what the expression tends towards, which helps in understanding its eventual behavior better.

Simplifying Expressions

Simplifying expressions in limits is an essential step to make evaluating them simpler. In calculus, you may encounter complex rational expressions where simplification becomes necessary for easy analysis.

• The goal is to reduce the expression to a form where you can easily see what happens as the variable increases or decreases indefinitely.
• In the given problem \( \lim_{x \to \infty} \frac{1}{3x + 6} \), we simplify by dividing both the numerator \( 1 \) and the denominator \( 3x + 6 \) by \( x \).
• This gives \( \frac{1/x}{3 + 6/x} \), which breaks down the main elements influencing the expression at large \( x \).

This process gives a clearer view of how each term behaves as \( x \) becomes very large, aiding in the evaluation process.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus. It involves determining what value a function approaches as the variable gets infinitely large or small. This requires a systematic approach to dealing with each part of the expression.

• Begin with expressing the limit problem clearly, observing which terms dominate the expression as the variable grows or decreases without bound.
• Next, simplify the expression as seen in prior steps - here in \( \frac{1/x}{3 + 6/x} \).
• As \( x \to \infty \), analyze the limiting behavior of each term: \( \frac{1}{x} \to 0 \) and \( \frac{6}{x} \to 0 \).

Finally, substitute these results into the simplified expression to find the limit. In our problem, \( \frac{0}{3 + 0} \) simplifies directly to \( 0 \), indicating that the original expression \( \lim_{x \to \infty} \frac{1}{3x + 6} \) equals 0. Understanding each step leads to confident and accurate evaluation of limits.