Find $\lim _{x \rightarrow-3} \frac{\sqrt{4 x+28}-4}{x+3}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{2}$ $\newline$ (B) $1$ $\newline$ (C) $2$ $\newline$ (D) The limit doesn't exist

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Algebra 1

Is (x, y) a solution to the system of equations?

Full solution

Q. Find

\lim _{x \rightarrow-3} \frac{\sqrt{4 x+28}-4}{x+3}

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{2}

\newline

(B)

1

\newline

(C)

2

\newline

(D) The limit doesn't exist

Identify the form: Identify the form of the limit. $\newline$ We need to find the limit of the function $(\sqrt{4x+28}-4)/(x+3)$ as $x$ approaches $-3$ . Let's first substitute $x = -3$ into the function to see if the limit can be directly calculated. $\newline$ $\lim_{x \to -3}(\sqrt{4x+28}-4)/(x+3) = (\sqrt{4(-3)+28}-4)/((-3)+3)$ $\newline$ $= (\sqrt{-12+28}-4)/0$ $\newline$ $= (\sqrt{16}-4)/0$ $\newline$ $= (4-4)/0$ $\newline$ $= 0/0$ $\newline$ This is an indeterminate form, so we cannot directly calculate the limit. We need to use algebraic manipulation to simplify the expression.
Apply algebraic manipulation: Apply algebraic manipulation to simplify the expression. $\newline$ To eliminate the indeterminate form, we can multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of $\sqrt{4x+28}-4$ is $\sqrt{4x+28}+4$ . $\newline$ $\lim_{x \to -3}\frac{\sqrt{4x+28}-4}{x+3} \cdot \frac{\sqrt{4x+28}+4}{\sqrt{4x+28}+4}$
Perform the multiplication: Perform the multiplication. $\newline$ Now, we multiply the numerators and the denominators separately. $\newline$ Numerator: $(\sqrt{4x+28}-4)(\sqrt{4x+28}+4) = (4x+28) - 16$ $\newline$ Denominator: $(x+3)(\sqrt{4x+28}+4)$
Simplify the expressions: Simplify the resulting expressions. $\newline$ Simplify the numerator: $\newline$ $(4x+28) - 16 = 4x + 12$ $\newline$ Simplify the denominator: $\newline$ We leave it as is for now: $(x+3)(\sqrt{4x+28}+4)$ $\newline$ Now the limit expression looks like this: $\newline$ $\lim_{x \to -3}\frac{4x + 12}{(x+3)(\sqrt{4x+28}+4)}$
Factor out common terms: Factor out common terms. $\newline$ We notice that the numerator has a term $4x + 12$ which can be factored as $4(x + 3)$ . $\newline$ $\lim_{x \to -3}\frac{4(x + 3)}{(x+3)(\sqrt{4x+28}+4)}$ $\newline$ Now we can cancel out the $(x+3)$ term in the numerator and denominator, as long as $x \neq -3$ . $\newline$ $\lim_{x \to -3}\frac{4}{(\sqrt{4x+28}+4)}$
Evaluate the limit: Evaluate the limit. $\newline$ Now that the expression is simplified, we can substitute $x = -3$ to find the limit. $\newline$ $\lim_{x \to -3}\frac{4}{(\sqrt{4x+28}+4)} = \frac{4}{(\sqrt{4(-3)+28}+4)}$ $\newline$ $= \frac{4}{(\sqrt{-12+28}+4)}$ $\newline$ $= \frac{4}{(\sqrt{16}+4)}$ $\newline$ $= \frac{4}{4+4}$ $\newline$ $= \frac{4}{8}$ $\newline$ $= \frac{1}{2}$