Problem 18 - Brief Applied Calculus

Question

Find the following limits without using a graphing calculator or making tables. $$ \lim _{q \rightarrow 9} \frac{8+2 \sqrt{q}}{8-2 \sqrt{q}} $$

Step-by-Step Solution

Answer

The limit is 7.

Step 1: Substitute the limit point into the expression

First, substitute \( q = 9 \) into the function \( \frac{8+2\sqrt{q}}{8-2\sqrt{q}} \). This gives us \( \frac{8+2\sqrt{9}}{8-2\sqrt{9}} = \frac{8+2\times3}{8-2\times3} = \frac{8+6}{8-6} = \frac{14}{2} \).

Step 2: Simplify the result

Simplify the fraction \( \frac{14}{2} \) by dividing the numerator by the denominator. This gives \( 7 \).

Key Concepts

Substitution Method

The substitution method is often the first step in finding limits in calculus. This method involves directly replacing the variable in the expression with the value to which it approaches. It's a straightforward approach for solving limit problems and usually provides quick results.

For the given exercise, the problem is to find the limit as the variable \( q \) approaches 9. The original expression is \[\frac{8 + 2 \sqrt{q}}{8 - 2 \sqrt{q}}\]The substitution method involves replacing \( q \) with 9, resulting in \( 2\sqrt{9} \). Consequently, the expression transforms into:

• Numerator: \( 8 + 2\times 3 = 14 \)
• Denominator: \( 8 - 2\times 3 = 2 \)

Plugging in these numbers gives a new fraction \( \frac{14}{2} \). This direct substitution shows that simplifying the resulting fraction will give us the value of the limit.

Simplifying Expressions

Simplifying expressions is an essential skill in calculus and helps make complex calculations more manageable. After substituting the point of approach into an expression, it's crucial to simplify the resulting fractions, radicals, or polynomials for clarity and ease of interpretation.

In this exercise, we simplified the expression \( \frac{14}{2} \) obtained after using the substitution method. Simplifying involves dividing the numerator by the denominator. In this case, that process is straightforward:

• Divide 14 by 2, resulting in 7.

Thus, the limit of the expression is 7. Simplification not only provides cleaner solutions but also aids in verifying the calculation steps, ensuring the accuracy of the limits obtained.

Rational Functions

Rational functions are a class of functions that are expressed as the ratio of two polynomials. They are prevalent in calculus and often require finding limits as part of their analysis. This exercise involved the rational function \[\frac{8 + 2 \sqrt{q}}{8 - 2 \sqrt{q}}\]As \( q \) approaches 9, the variable is replaced, the expression still retains its rational form of a numerator divided by a denominator.

Handling limits of rational functions typically involves checking both the numerator and the denominator. Direct substitution is one method to handle these types of problems, and if it results in a determinate form such as \( \frac{a}{b}\) where \( b eq 0 \), then simplifying will directly yield the limit.

• Identify the roles of polynomials in the numerator and denominator.
• Substitute the point into the function and simplify.

Understanding rational functions and their behavior at limits helps in analyzing how the function behaves as the variable approaches a specific point.