Find $\lim _{x \rightarrow 4} \frac{x-4}{3-\sqrt{x+5}}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-6$ $\newline$ (B) $-\frac{4}{3}$ $\newline$ (C) $-\frac{10}{3}$ $\newline$ D The limit doesn't exist

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Q. Find

\lim _{x \rightarrow 4} \frac{x-4}{3-\sqrt{x+5}}

\newline

Choose

1

answer:

\newline

(A)

-6

\newline

(B)

-\frac{4}{3}

\newline

(C)

-\frac{10}{3}

\newline

D The limit doesn't exist

Identify Limit Form: Identify the form of the limit as $x$ approaches $4$ . We need to check if the limit results in an indeterminate form by substituting $x = 4$ into the expression $(x-4)/(3-\sqrt{x+5})$ . Calculate the numerator and denominator separately: Numerator: $x - 4 = 4 - 4 = 0$ Denominator: $3 - \sqrt{x + 5} = 3 - \sqrt{4 + 5} = 3 - \sqrt{9} = 3 - 3 = 0$ Since both the numerator and denominator are $0$ , we have an indeterminate form of $0/0$ .
Check Indeterminate Form: Simplify the expression to resolve the indeterminate form. $\newline$ We can use algebraic manipulation to simplify the expression. One common technique is to multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ The conjugate of the denominator $3 - \sqrt{x + 5}$ is $3 + \sqrt{x + 5}$ . $\newline$ Multiply the original expression by $(3 + \sqrt{x + 5})/(3 + \sqrt{x + 5})$ : $\newline$ $\lim_{x \rightarrow 4} \frac{(x-4)(3 + \sqrt{x + 5})}{(3 - \sqrt{x + 5})(3 + \sqrt{x + 5})}$
Simplify Expression: Perform the multiplication in the numerator and use the difference of squares in the denominator. $\newline$ Numerator: $(x - 4)(3 + \sqrt{x + 5})$ $\newline$ Denominator: $(3 - \sqrt{x + 5})(3 + \sqrt{x + 5}) = 3^2 - (\sqrt{x + 5})^2$ $\newline$ Now, simplify the denominator: $\newline$ Denominator: $9 - (x + 5) = 9 - x - 5 = 4 - x$
Perform Multiplication: Notice that the numerator can be simplified further. $\newline$ We can expand the numerator: $\newline$ Numerator: $x \times 3 + x \times \sqrt{x + 5} - 4 \times 3 - 4 \times \sqrt{x + 5}$ $\newline$ Simplify the numerator: $\newline$ Numerator: $3x + x \times \sqrt{x + 5} - 12 - 4 \times \sqrt{x + 5}$
Simplify Numerator: Cancel out the common terms in the numerator and denominator. $\newline$ We have a common term of $x - 4$ in the numerator and $4 - x$ in the denominator, which can be simplified to $-1$ when we reverse the order of the terms in the denominator. $\newline$ So, the expression simplifies to: $\newline$ $\lim_{x \to 4} \frac{-1(3 + \sqrt{x + 5})}{-1}$ $\newline$ The $-1$ in the numerator and denominator cancel out, leaving us with: $\newline$ $\lim_{x \to 4} (3 + \sqrt{x + 5})$
Cancel Common Terms: Evaluate the limit by substituting $x = 4$ into the simplified expression. $\newline$ $\lim_{x \to 4} (3 + \sqrt{x + 5}) = 3 + \sqrt{4 + 5} = 3 + \sqrt{9} = 3 + 3 = 6$ $\newline$ However, we must remember that we canceled a negative sign in Step $5$ , so the actual limit is the negative of this result: $\newline$ Final limit: $-6$