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Magazine Chapter 3: Problem 45
Find the limits graphically. Then confirm algebraically. $$ \lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-2} $$
Short Answer
Expert verified
The limit is 4.
Step by step solution
01
Understand the Problem
We need to find the limit of the function \( f(x) = \frac{x - 4}{\sqrt{x} - 2} \) as \( x \) approaches 4. This function may have an indeterminate form if we directly substitute \( x = 4 \). We'll first analyze the behavior of the function graphically and then confirm it algebraically.
02
Graphical Analysis
Plot the function \( f(x) = \frac{x - 4}{\sqrt{x} - 2} \) around \( x = 4 \). Observing the graph will help visualize the behavior of the function near \( x = 4 \). You'll likely notice that as \( x \) approaches 4 from both sides, the value of \( f(x) \) seems to approach a particular number.
03
Simplify the Expression
Rewrite \( \frac{x - 4}{\sqrt{x} - 2} \) using algebraic manipulation. Start by multiplying the numerator and the denominator by the conjugate of the denominator: \( \sqrt{x} + 2 \). This gives \( \frac{(x - 4)(\sqrt{x} + 2)}{(\sqrt{x} - 2)(\sqrt{x} + 2)} \).
04
Factor and Cancel Terms
The denominator simplifies due to the difference of squares: \( (\sqrt{x} - 2)(\sqrt{x} + 2) = x - 4 \). Therefore, the expression becomes \( \frac{(x - 4)(\sqrt{x} + 2)}{x - 4} \). Since \( x eq 4 \) in the limit, we can cancel \( x - 4 \), simplifying the expression to \( \sqrt{x} + 2 \).
05
Evaluate the Limit
Now the limit is straightforward. Substitute \( x = 4 \) into \( \sqrt{x} + 2 \) to find the limit. Therefore, \( \lim_{x \rightarrow 4} (\sqrt{x} + 2) = \sqrt{4} + 2 = 2 + 2 = 4 \).
06
Confirm with Graph
Compare the computed algebraic limit with the graph's value at \( x = 4 \). The graph should confirm that as \( x \) approaches 4 from both sides, \( f(x) \) approaches 4.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphical Analysis
Graphical analysis is an intuitive way to understand how a function behaves as the input approaches a particular value. By plotting the function, you can visually assess how the function behaves near the limit, providing a tangible insight into the problem.
For the expression \( \frac{x - 4}{\sqrt{x} - 2} \), plotting it around \( x = 4 \) will showcase the function's behavior as \( x \) gets close to 4.
For the expression \( \frac{x - 4}{\sqrt{x} - 2} \), plotting it around \( x = 4 \) will showcase the function's behavior as \( x \) gets close to 4.
- Start by graphing the function in a range that includes \( x = 4 \).
- Observe how the function moves as \( x \) moves closer to 4 from both the left and the right.
- The plot will typically show that as \( x \to 4 \), \( f(x) \) gravitates toward a specific value.
Algebraic Confirmation
Algebraic confirmation involves manipulating the function to remove indeterminate forms and confirm results graphically suggested.
When faced with indeterminate forms like \( \frac{x - 4}{\sqrt{x} - 2} \), algebraic techniques ensure that the limit equals a specific value.
When faced with indeterminate forms like \( \frac{x - 4}{\sqrt{x} - 2} \), algebraic techniques ensure that the limit equals a specific value.
- Simplifying the Expression: Start by rewriting the expression with algebraic manipulation.
- Factor and Simplify: Often, rearranging or simplifying the function can reveal the limit.
Indeterminate Forms
Indeterminate forms occur when substitution of the limit value into a function's expression leads to undefined or ambiguous results, such as \( \frac{0}{0} \). This does not necessarily mean there is no limit; rather, it needs a different method for evaluation.
In the exercise \( \lim_{x \rightarrow 4} \frac{x - 4}{\sqrt{x} - 2} \), direct substitution of \( x = 4 \) gives \( \frac{0}{0} \), indicating an indeterminate form.
Consider using conjugates, factoring, or L'H么pital's rule as algebraic techniques to resolve these forms:
In the exercise \( \lim_{x \rightarrow 4} \frac{x - 4}{\sqrt{x} - 2} \), direct substitution of \( x = 4 \) gives \( \frac{0}{0} \), indicating an indeterminate form.
Consider using conjugates, factoring, or L'H么pital's rule as algebraic techniques to resolve these forms:
- Multiplication by the conjugate is a powerful, often-used method.
- Factoring out common terms can help make limits more apparent.
- L'H么pital's Rule offers alternatives through derivative computation.
Algebraic Manipulation
Algebraic manipulation involves reconfiguring a mathematical expression to simplify a problem or reveal solutions. It's especially useful in calculus for resolving limits.
For the exercise, rewrite \( \frac{x - 4}{\sqrt{x} - 2} \) by multiplying numerator and denominator by the conjugate \( \sqrt{x} + 2 \). This transforms the denominator using the difference of squares formula:
By simplifying the expression, you transform an indeterminate form into a computable solution, confirming limits both visually and algebraically.
For the exercise, rewrite \( \frac{x - 4}{\sqrt{x} - 2} \) by multiplying numerator and denominator by the conjugate \( \sqrt{x} + 2 \). This transforms the denominator using the difference of squares formula:
- The denominator becomes \( x - 4 \).
- Cancel \( x - 4 \) in the expression since it doesn't equal zero as \( x \to 4 \).
By simplifying the expression, you transform an indeterminate form into a computable solution, confirming limits both visually and algebraically.
