Question

Evaluate the given limit. $$ \lim _{x \rightarrow-2} \frac{x^{2}-5 x-14}{x^{2}+10 x+16} $$

Short answer

The limit is \(-\frac{3}{2}\).

Step-by-step solution

Identify Limit Type

Examine both the numerator and denominator as \(x\) approaches \(-2\). Plug it in briefly to determine form of limit. If plugging in \(-2\) gives \(\frac{0}{0}\), factor further.

Factor the Numerator and Denominator

Factor the numerator \(x^2 - 5x - 14\) into \((x-7)(x+2)\). Factor the denominator \(x^2 + 10x + 16\) into \((x+2)(x+8)\).

Simplify the Expression

After factoring, cancel the common factor \((x+2)\) from both numerator and denominator, obtaining \(\frac{x-7}{x+8}\).

Evaluate the Simplified Limit

Substitute \(x = -2\) into the simplified expression, \(\frac{x-7}{x+8}\), yielding \(\frac{-2-7}{-2+8} = \frac{-9}{6} = -\frac{3}{2}\).

Key concepts

Factoring Polynomials

Polynomials can be complex but factoring makes them simpler. Factoring them involves writing them as a product of smaller simpler polynomials. It's a tool used in algebra to break down expressions for easier manipulation. When you evaluate limits like the one given in the exercise, factoring helps you overcome the indeterminate forms, such as \(\frac{0}{0}\). When you see a polynomial like \(x^2 - 5x - 14\), you want to express it in the form \((x-a)(x-b)\). To do this, look for two numbers that multiply to -14 and add to -5. For this polynomial, those numbers are -7 and 2, giving us the factors \((x-7)(x+2)\). The denominator \(x^2 + 10x + 16\) is factored similarly, using the numbers 2 and 8 which give \((x+2)(x+8)\). Factoring allows us to simplify expressions and solve equations more easily, especially when dealing with limits.

Rational Functions

Rational functions are expressions that are the ratio of two polynomials. They have the general form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. When evaluating the limits of rational functions, especially at specific points, division by zero can often be a problem. To manage this, we factor each polynomial and simplify. In our exercise, the function \(\frac{x^2-5x-14}{x^2+10x+16}\) was simplified by canceling out the common factors of \((x+2)\) from both numerator and denominator. This makes it manageable to evaluate the limit at \(x = -2\), which initially resulted in an indeterminate form. Rational functions appear in various real-world applications such as calculating rates, percentages, or even modeling certain natural processes. Understanding how to manipulate them effectively is a critical math skill.

Indeterminate Form

In calculus, the indeterminate form \(\frac{0}{0}\) often arises during limit evaluation. This form occurs when both the numerator and denominator of a rational function tend to zero. However, this doesn't mean the limit does not exist. Instead, it usually requires deeper analysis to resolve. In the given problem, after substituting \(x = -2\) initially, both the numerator and the denominator became zero, which led to an indeterminate form. The next steps are to factor polynomials and simplify them to check if the limit can be resolved. Fortunately, by canceling the common term \((x+2)\), we simplified the expression to a form that can be evaluated at \(x = -2\) without leading to an undefined result. Handling indeterminate forms is key in calculus. It ensures you can logically approach and solve limit problems that appear unsolvable at a first glance.