Example 4 Evaluate lim_(x rarr4)(x^(2)-16)/(x-4).

Recognize Problem: We are given the limit expression: lim_(x -> 4)(x^2 - 16)/(x - 4) First, we need to recognize that direct substitution of x = 4 into the expression would result in a division by zero, which is undefined. Therefore, we need to simplify the expression before substituting the value of x.Factor Numerator: Factor the numerator. The numerator x^2 - 16 is a difference of squares and can be factored as: x^2 - 16 = (x + 4)(x - 4)Simplify Expression: Simplify the expression. Now we can cancel out the common factor (x - 4) from the numerator and the denominator: (x^2 - 16)/(x - 4) = ((x + 4)(x - 4))/(x - 4) After canceling, we get: (x + 4)Substitute Value: Substitute the value of x. Now that the expression is simplified and the problematic factor is removed, we can substitute x = 4 into the expression: lim_(x -> 4)(x + 4) = 4 + 4Calculate Final Value: Calculate the final value. After substitution, we find that: 4 + 4 = 8 So, the limit as x approaches 4 of the expression (x^2 - 16)/(x - 4) is 8.

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Algebra 2

Evaluate expression when two complex numbers are given

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Q. Example

4

Evaluate

\lim _{x \rightarrow 4} \frac{x^{2}-16}{x-4}

Recognize Problem: We are given the limit expression: $\newline$ $\lim_{x \to 4}\frac{x^2 - 16}{x - 4}$ $\newline$ First, we need to recognize that direct substitution of $x = 4$ into the expression would result in a division by zero, which is undefined. Therefore, we need to simplify the expression before substituting the value of $x$ .
Factor Numerator: Factor the numerator. $\newline$ The numerator $x^2 - 16$ is a difference of squares and can be factored as: $\newline$ $x^2 - 16 = (x + 4)(x - 4)$
Simplify Expression: Simplify the expression. $\newline$ Now we can cancel out the common factor $(x - 4)$ from the numerator and the denominator: $\newline$ $(x^2 - 16)/(x - 4) = ((x + 4)(x - 4))/(x - 4)$ $\newline$ After canceling, we get: $\newline$ $(x + 4)$
Substitute Value: Substitute the value of $x$ . Now that the expression is simplified and the problematic factor is removed, we can substitute $x = 4$ into the expression: $\lim_{x \to 4}(x + 4) = 4 + 4$
Calculate Final Value: Calculate the final value. $\newline$ After substitution, we find that: $\newline$ $4 + 4 = 8$ $\newline$ So, the limit as $x$ approaches $4$ of the expression $\frac{x^2 - 16}{x - 4}$ is $8$ .