Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit, if it exists. $$\lim _{x \rightarrow 1} 9^{1 /(x+1)}$$

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Understand the Limit Problem

We need to find the limit of the expression \( 9^{1/(x+1)} \) as \( x \) approaches 1. This involves assessing how this expression behaves when \( x \) gets very close to 1.

Step 2: Substitute the Limit Value

Begin by substituting \( x = 1 \) into the expression. The expression becomes \( 9^{1/(1+1)} = 9^{1/2} \).

Step 3: Simplify the Expression

Simplify \( 9^{1/2} \), which is the square root of 9. Thus, \( \sqrt{9} = 3 \).

Step 4: State the Limit Result

Since after substituting and simplifying we found that \( 9^{1/2} = 3 \), the limit of the expression \( 9^{1/(x+1)} \) as \( x \) approaches 1 is 3.

Key Concepts

Limit Evaluation

Limit evaluation is a fundamental concept in calculus. It involves finding the value that a function approaches as the input variable gets arbitrarily close to a given point. This helps us understand the behavior of functions near specific points, even when the function might not be defined at that point itself. When approaching this, imagine a car getting closer and closer to a red traffic light. The limit is the point it intends to stop.

In the original exercise, we needed to find the limit as \( x \) approaches 1 for the expression \( 9^{1/(x+1)} \). This means we are interested in seeing what value the expression is getting closer to as \( x \) nudges closer to 1.

• Check the environment around the point: We examined the behavior when \( x \) is getting near 1.
• Observe the expression's tendency: This provides valuable intuition into the function’s behavior around that point.

Evaluating limits can handle scenarios where direct substitution might cause confusion or seem undefined at first glance, providing a clearer picture of the function’s behavior.

Substitution Method

The substitution method is often the first step in evaluating limits. It simply means replacing the input variable with its limiting value to see if we can directly compute the limit. This approach works well when the function is continuous at the point.

In the example provided, we began by substituting \( x = 1 \) into the expression \( 9^{1/(x+1)} \). It transformed the expression into \( 9^{1/2} \).

• This method is straightforward and often pays off immediately when functions are well-behaved at the limits.
• Observe the constants being derived after substitution for simpler computation.

This method is an excellent first approach to try with limits, as it quickly tells you if further steps are necessary, especially when the expression cleanly resolves after substitution.

Simplifying Expressions

Simplifying expressions is a crucial skill in evaluating limits because simplified expressions are easier to handle and compute. It often involves reducing a complex expression to its most basic form.

In the original exercise, after substituting \( x = 1 \), we simplified \( 9^{1/2} \). This expression is another way to represent the square root of 9. We found that \( \sqrt{9} = 3 \).

• Breaking down powers and roots in expressions can lead to simpler computation.
• Always work towards reducing the expression to its simplest form for easy evaluation.

Regularly practicing simplification can significantly improve your ability to evaluate limits efficiently. It's like organizing your room; the goal is to find what you need quickly and accurately.