Problem 6 - Technical Mathematics with Calculus

Question

Evaluate each limit. $$\lim _{x \rightarrow-7} \frac{49-x^{2}}{x+7}$$

Step-by-Step Solution

Answer

\( \lim_{x \rightarrow -7} \frac{49 - x^2}{x + 7} = 14 \)

Step 1: Factor the Numerator

First, factor the quadratic expression in the numerator. The expression \(49 - x^2\) is a difference of squares and can be factored into \( (7 + x)(7 - x) \).

Step 2: Simplify the Fraction

Next, simplify the fraction by canceling out the common factor of \(x + 7\) in the numerator and the denominator. The simplification yields \(7-x\).

Step 3: Substitute the Limit Value

Substitute \(x = -7\) into the simplified expression \(7-x\) to evaluate the limit.

Key Concepts

Difference of Squares

When evaluating limits involving quadratic expressions, one may encounter the 'difference of squares', which is a specific type of algebraic identity. This identity takes the form of \( a^2 - b^2 \) and can be factored into \( (a + b)(a - b) \). In the context of limits, factoring a difference of squares is useful because it can simplify the algebraic expression and potentially eliminate indeterminate forms that make direct substitution impossible.

For example, the expression \( 49 - x^2 \) is a difference of squares because it resembles the form \( a^2 - b^2 \) where \( a = 7 \) and \( b = x \). Therefore, it can be factored into \( (7 + x)(7 - x) \), thus simplifying the equation further and preparing it for the next steps of solving a limit.

Factoring Quadratics

Factoring quadratics is a fundamental skill for algebra, precalculus, and calculus students. A quadratic expression is traditionally in the form \( ax^2 + bx + c \). Factoring involves rewriting this expression as a product of two binomials. This technique is crucial when working with limits in calculus since it can transform an expression into a simpler form that allows for direct substitution or further simplification.

In the provided exercise, the numerator is a special case of a quadratic known as a difference of squares. Factoring it correctly is essential for simplifying the expression and progressing towards evaluating the limit. One common pitfall for students to avoid when factoring is incorrect sign usage or misidentifying the square terms. That's why this step must be approached carefully and methodically, ensuring that the factored form accurately reflects the original quadratic expression.

Limit Simplification

Limit simplification is a process that often includes several calculus techniques to make a complex expression more approachable. When a direct substitution in a limit leads to a form like \( 0/0 \) or \( \infty/\infty \)—known as indeterminate forms—it suggests that further simplification is necessary to find the limit's value. One effective technique for simplification is factoring, as seen with the difference of squares in quadratics. Another is canceling common factors in the numerator and denominator.

After canceling, the limit may become straightforward enough to evaluate through direct substitution. As in our example, once the common factor \( x + 7 \) is canceled from the numerator and denominator, the expression simplifies, and the limit becomes clear. This is an excellent demonstration of how decomposing and restructuring a complex form can reveal simple, solvable problems. Understanding these algebraic manipulations is just as crucial as understanding the underlying calculus concepts.