Problem 25 - Brief Applied Calculus

Question

Find the following limits without using a graphing calculator or making tables. $$ \lim _{x \rightarrow-3} \frac{x+3}{x^{2}+8 x+15} $$

Step-by-Step Solution

Answer

The limit is \(\frac{1}{2}\).

Step 1: Identify the Limit Type

The expression provided is a rational function \[\frac{x+3}{x^2+8x+15}\]. Our task is to find out if directly substituting the value \(x = -3\) leads to any indeterminate form.

Step 2: Check for Direct Substitution

By directly substituting \(x = -3\) into the function, we calculate both the numerator and the denominator. The numerator becomes \((-3) + 3 = 0\) and the denominator becomes \((-3)^2 + 8(-3) + 15 = 9 - 24 + 15 = 0\). This results in an indeterminate form \(\frac{0}{0}\).

Step 3: Factor the Denominator

We need to factor the quadratic expression in the denominator \(x^2 + 8x + 15\). The factors of 15 that add up to 8 are 3 and 5. Thus, \(x^2 + 8x + 15 = (x + 3)(x + 5)\).

Step 4: Simplify the Expression

After factoring, the expression becomes \(\frac{x+3}{(x+3)(x+5)}\). The common factor \(x + 3\) can be canceled from the numerator and denominator, simplifying the expression to \(\frac{1}{x+5}\) for \(x eq -3\).

Step 5: Take the Limit of the Simplified Expression

Now, find the limit of the simplified expression \(\frac{1}{x+5}\) as \(x\) approaches \(-3\). Substitute the value \(-3\) into the simplified expression to get \(\frac{1}{-3+5} = \frac{1}{2}\).

Step 6: Conclusion

The limit of \(\frac{x+3}{x^2+8x+15}\) as \(x\) approaches \(-3\) is \(\frac{1}{2}\).

Key Concepts

Rational Functions

Rational functions are a crucial part of calculus, especially when dealing with limits. A rational function is any function that can be expressed as the ratio of two polynomials. In this context, our function is \( \frac{x+3}{x^2+8x+15} \). Each polynomial, whether in the numerator or the denominator, determines the behavior of the function.

• The degree of the polynomial in the numerator and denominator affects the values the rational function can take.
• Rational functions can have vertical asymptotes, holes, or horizontal asymptotes based on the roots of these polynomials.

These functions provide insight into behavior beyond mere numbers. When assessing limits, particularly with a value approaching a certain point, understanding the structure of these polynomials helps identify potential zeros or undefined points. This is the first step in determining whether manipulating the function could simplify finding a limit.

Indeterminate Forms

In calculus, indeterminate forms often appear when evaluating limits, especially for rational functions. When we directly substitute \(x = -3\) in the expression \(\frac{x+3}{x^2+8x+15}\), it results in \(\frac{0}{0}\). This is a classic indeterminate form. What does this mean?

• Indeterminate forms indicate that the usual methods for finding a limit, like direct substitution, won’t work.
• This form suggests that there is more algebraic manipulation that can be done to further simplify the function.

Once we recognize an indeterminate form, the next step is to either factor or find another method to simplify the function. Canceling common factors is a common technique used to resolve indeterminate forms.

Factoring Polynomials

Factoring is a fundamental technique used both in algebra and calculus. It involves writing a polynomial as the product of its linear factors, which often simplifies complex expressions. In the problem \(x^2 + 8x + 15\), we see this quadratic equation can be factored as \((x+3)(x+5)\).

• Factoring helps identify roots or zeros of the polynomial, which are points where the function might be undefined.
• This step is particularly useful for simplifying rational expressions and resolving indeterminate forms.

Once factors are identified, common terms in the numerator and the denominator can be canceled out, making the calculation of limits feasible. Factoring transforms complex calculations into manageable parts, thus ensuring the correct application of limit theorems.