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Magazine Chapter 13: Problem 38
Determine each limit, if it exists. $$\lim _{x \rightarrow 3} 5^{\sqrt{x+1}}$$
Short Answer
Expert verified
The limit is 25.
Step by step solution
01
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.
02
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: \( \sqrt{3+1} = \sqrt{4} = 2 \).
03
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: \( 5^2 = 25 \).
04
Conclusion
Thus, the limit of the function \( 5^{\sqrt{x+1}} \) as \( x \) approaches 3 is 25.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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.
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:
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.
