Problem 42 - Thomas Calculus

Question

Limits of quotients Find the limits in Exercises \(23-42\) $$\lim _{x \rightarrow 4} \frac{4-x}{5-\sqrt{x^{2}+9}}$$

Step-by-Step Solution

Answer

The limit is \( \frac{5}{4} \).

Step 1: Identify the Indeterminate Form

First, substitute \( x = 4 \) into the expression \( \frac{4-x}{5-\sqrt{x^2+9}} \).\[\frac{4-4}{5-\sqrt{4^2+9}} = \frac{0}{0}\]This yields the indeterminate form \( \frac{0}{0} \), indicating that we need to simplify the expression further.

Step 2: Rationalize the Denominator

To eliminate the square root in the denominator, multiply the numerator and the denominator by the conjugate of the denominator:\[\frac{4-x}{5-\sqrt{x^2+9}} \times \frac{5+\sqrt{x^2+9}}{5+\sqrt{x^2+9}} = \frac{(4-x)(5+\sqrt{x^2+9})}{(5-\sqrt{x^2+9})(5+\sqrt{x^2+9})}\]The denominator becomes a difference of squares:\[(5-\sqrt{x^2+9})(5+\sqrt{x^2+9}) = 25 - (x^2 + 9) = 16 - x^2\]

Step 3: Simplify the Expression

Now simplify the expression. Substitute the simplified denominator back into the expression:\[\frac{(4-x)(5+\sqrt{x^2+9})}{16-x^2}\]Factor the denominator as a difference of squares:\[16 - x^2 = (4-x)(4+x)\]

Step 4: Cancel Common Terms

Cancel the common terms \((4-x)\) from the numerator and the denominator:\[\frac{5+\sqrt{x^2+9}}{4+x}\]

Step 5: Evaluate the Limit

Substitute \( x = 4 \) into the simplified expression:\[\lim_{x \to 4} \frac{5+\sqrt{x^2+9}}{4+x} = \frac{5+\sqrt{4^2+9}}{4+4}\]Simplify the expression:\[= \frac{5+\sqrt{25}}{8} = \frac{5+5}{8} = \frac{10}{8} = \frac{5}{4}\]

Step 6: Conclusion: State the Limit

The limit of the given expression as \( x \) approaches 4 is \( \frac{5}{4} \).

Key Concepts

Indeterminate Form

When attempting to find the limit of a function as a variable approaches a specific value, you may encounter an indeterminate form. This occurs when substituting the value into the function results in expressions like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), or \( 0 \cdot \infty \), among others. In our problem, substituting \( x = 4 \) into the expression \( \frac{4-x}{5-\sqrt{x^2+9}} \) creates an indeterminate form of \( \frac{0}{0} \).

• The indeterminate form tells us that the limit cannot be easily evaluated by straightforward substitution.
• It signals a need for algebraic manipulation or simplification to properly evaluate the limit.

Understanding indeterminate forms is crucial for accurate limit evaluation in calculus, providing the foundation for simplifying complex problems.

Rationalization Technique

Rationalization is a powerful algebraic technique used to simplify expressions involving radicals. It is often employed to resolve indeterminate forms by eliminating square roots or other complex terms in the denominator. In the exercise, we tackled the indeterminate form of \( \frac{0}{0} \) by rationalizing the denominator of \( \frac{4-x}{5-\sqrt{x^2+9}} \).

• We multiplied both the numerator and denominator by the conjugate \( 5+\sqrt{x^2+9} \).
• This step creates a difference of squares in the denominator, simplifying it to \( 16-x^2 \).
• Rationalization helps bridge the gap to further simplify or evaluate limits accurately.

This technique can be a critical tool in calculus for transforming complex rational expressions into solvable forms.

Limit Evaluation

Evaluating limits is a fundamental skill in calculus. After rationalizing the equation, you often simplify further to find the limit. Starting with our conjugate modification from the previous step, we found a simpler expression: \( \frac{(4-x)(5+\sqrt{x^2+9})}{16-x^2} \).

• Next, noticing that \( 16-x^2 \) can be factored into \( (4-x)(4+x) \) aids in canceling common terms: \( (4-x) \).
• With simplification, the function reduces to \( \frac{5+\sqrt{x^2+9}}{4+x} \).
• To find the limit, substitute \( x = 4 \) into this simplified form to avoid indeterminacy and achieve \( \frac{5}{4} \).

Proper limit evaluation extends beyond mere substitution and requires strategic manipulation to avoid indeterminate forms.

Calculus Problem Solving

Solving calculus problems involves a combination of techniques such as recognizing indeterminate forms, employing algebraic manipulation, and strategically simplifying expressions. To resolve the limit \( \lim_{x \rightarrow 4} \frac{4-x}{5-\sqrt{x^2+9}} \):

• Identify potential obstacles like indeterminate forms.
• Apply appropriate algebraic strategies such as rationalization and simplification.
• Carefully substitute values in the transformed expression to accurately determine limits.

In essence, calculus problem solving is about using logical, methodical steps to navigate complex mathematical scenarios effectively. Mastery of these techniques helps you tackle a wide range of calculus problems with confidence.