Question

Evaluate each limit. $$\lim _{x \rightarrow 4} \frac{x^{2}-16}{x-4}$$

Short answer

The value of the limit is 8.

Step-by-step solution

Factor the numerator

Factor the expression in the numerator to identify common factors with the denominator. The difference of squares factoring technique can be applied: \(x^2 - 16 = (x - 4)(x + 4)\).

Simplify the fraction

Cancel out the common factor \((x - 4)\) from the numerator and denominator, which is allowed since we are considering the limit and \(x\) is approaching 4 but not equal to 4. This simplifies the expression to \(x + 4\).

Substitute the limiting value

Substitute the value that \(x\) is approaching into the simplified expression. When \(x = 4\), the expression evaluates to \(4 + 4\).

Calculate the final result

Simplify the expression to find the value of the limit: \(4 + 4 = 8\). Therefore, the limit of the function as \(x\) approaches 4 is 8.

Key concepts

Factoring Polynomials

Understanding how to factor polynomials is vital in calculus, especially when evaluating limits. Factoring involves rewriting a complex expression as a product of simpler ones. In the case of the given limit problem, we encounter a common algebraic form known as a difference of squares:

A difference of squares is identified by an expression like
\[ a^2 - b^2 \]
which factors into
\[ (a + b)(a - b) \].

This identity allows us to break down the numerator of our fraction, making it easier to simplify. For example, in our exercise, u
\[ x^2 - 16 \]
becomes
\[ (x + 4)(x - 4) \]
by recognizing that \( 16 \) is a perfect square, equivalent to \( 4^2 \).The technique is powerful when looking for common factors with the denominator, as it often leads to the cancellation of terms when simplifying expressions in limit problems.

Simplifying Expressions

Upon successfully factoring polynomials, the next step is simplifying expressions. This involves reducing expressions to their simplest form, which, in the context of limits, often includes canceling out common factors. For our exercise, once factored, the expression
\[ \frac{(x+4)(x-4)}{x-4} \]
has a visible common factor of \( (x-4) \) in both numerator and denominator.

Since we are dealing with a limit where \( x \) is not exactly \( 4 \), but only approaching it, we can safely cancel the \( (x-4) \) terms. The remaining expression \( x+4 \) is then the simplified form. Simplifying is a significant step because it often leads to a straightforward substitution in the following steps, making the limit much easier to evaluate.

Substituting Limit Values

The final perspective we take in solving limit problems is substituting limit values. In the given problem, after simplifying our expression to \( x+4 \), we substitute the limiting value of \( x \), which is 4 in this case.

It's important to clarify that when substituting, we are not implying that \( x \) equals 4, but rather that it's exceedingly close to 4. Substitution is what helps us understand the behavior of the function as it nears a particular value. It tells us, effectively, what the function's output is as the input approaches some value.

By substituting 4 into our simplified polynomial, we determine that the limit is \( 4+4 \) or \( 8 \). The process of substitution is crucial in calculus as it bridges the gap between abstract simplification and tangible results.