Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit, if it exists. $$\lim _{x \rightarrow 3} 5^{\sqrt{x+1}}$$

Step-by-Step Solution

Answer

The limit is 25.

Step 1: Understand the Limit Expression

We are given the limit expression \( \lim_{x \to 3} 5^{\sqrt{x+1}} \). This means as \( x \) approaches 3, we need to determine the value that \( 5^{\sqrt{x+1}} \) approaches.

Step 2: Substitute the Limit Value into the Inner Expression

First, we evaluate the inner expression \( \sqrt{x+1} \) as \( x \to 3 \). Substitute \( x = 3 \) into the inner expression: \( \sqrt{3+1} = \sqrt{4} = 2 \).

Step 3: Evaluate the Outer Expression

Next, substitute the result from Step 2 into the function \( 5^{\sqrt{x+1}} \). Since \( \sqrt{x+1} \to 2 \) as \( x \to 3 \), substitute 2 into the outer function: \( 5^2 = 25 \).

Step 4: Conclusion

Thus, the limit of the function \( 5^{\sqrt{x+1}} \) as \( x \) approaches 3 is 25.

Key Concepts

Evaluating Limits

Limits are a fundamental concept in calculus that help us understand the behavior of a function as its input approaches a specific value. Evaluating limits is essentially about predicting where a function is headed without actually reaching that point. Imagine you're following a road to a destination. Although you might not get there immediately, knowing the limits tells us what to expect as we get closer.

To evaluate limits, you might need to:

• Substitute the value into the function directly, if possible.
• If direct substitution results in an indeterminate form, such as 0/0, consider using algebraic simplification or other techniques.
• Use graphical or numerical methods for complex functions, ensuring continuity along the path to the limit point.

By carefully applying these methods, we can determine the limit, if it exists, and gain insights into the behavior of functions near specific points.

Square Root Function

A square root function is a type of function in mathematics that involves the square root of an expression. It is denoted as \( \sqrt{x} \), which represents a value that, when multiplied by itself, gives the number \( x \). In our exercise, we dealt with the square root of \( x+1 \).

Here are a few key points about square root functions:

• They are fundamental to understanding many algebraic problems as they reverse the operation of squaring numbers.
• The function \( \sqrt{x} \) only produces non-negative results, which is crucial when working within the domain of real numbers.
• The square root function grows at a decreasing rate, meaning it rises quickly for small values of \( x \), but more slowly as \( x \) increases.

When analyzing limits that involve square root functions, substituting and simplifying can often reveal the limit value, as seen when we substituted 3 into \( \sqrt{x+1} \) to find \( \sqrt{4} = 2 \). This simplification aids in the subsequent calculations of the problem at hand.

Exponential Functions

Exponential functions are powerful mathematical constructs characterized by their growth rates, defined as functions of the form \( a^x \), where \( a \) is a constant base and \( x \) is the exponent. In the given exercise, we encountered the exponential function \( 5^{\sqrt{x+1}} \), showcasing an exponent that is itself a function of \( x \).

Some important aspects of exponential functions include:

• They exhibit rapid growth or decay, depending on whether the base is greater than or less than 1.
• Exponential functions are used in many fields, such as population growth, radioactive decay, and finance for compounding interest.
• In terms of limits, evaluating an exponential function often involves manipulating the exponent to a more manageable form, as seen in our case when \( \sqrt{x+1} \) simplified to 2.

Thus, once we realized that the inner square root expression approached 2, calculating \( 5^2 = 25 \) was straightforward. This stepwise simplification highlights the efficiency of exponential functions when analyzing limits.