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Mandisa's teacher gave her a flow chart (below) and asked her to find
lim_(x rarr-1)f(x) for
f(x)=(1+sqrt(5x+30))/(x^(2)-1).
Calculating
lim_(x rarr a)f(x)

Mandisa's teacher gave her a flow chart (below) and asked her to find $\lim _{x \rightarrow-1} f(x)$ for $f(x)=\frac{1+\sqrt{5 x+30}}{x^{2}-1}$ . $\newline$ Calculating $\lim _{x \rightarrow a} f(x)$

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Find derivatives using the quotient rule II

Full solution

Q. Mandisa's teacher gave her a flow chart (below) and asked her to find

\lim _{x \rightarrow-1} f(x)

for

f(x)=\frac{1+\sqrt{5 x+30}}{x^{2}-1}

.

\newline

Calculating

\lim _{x \rightarrow a} f(x)

Check for Undefined: First, substitute $x = -1$ into the function to check if it's defined. $\newline$ $f(-1) = \frac{1 + \sqrt{5(-1) + 30}}{(-1)^2 - 1} = \frac{1 + \sqrt{25}}{1 - 1} = \frac{1 + 5}{0} = \frac{6}{0}$ $\newline$ Since the denominator is $0$ , the function is undefined at $x = -1$ .
Factor Denominator: Next, factor the denominator to see if we can simplify the function. $\newline$ $x^2 - 1 = (x - 1)(x + 1)$ $\newline$ So, $\newline$ $f(x) = \frac{1 + \sqrt{5x + 30}}{(x - 1)(x + 1)}$
Apply L'Hôpital's Rule: Now, let's use the limit properties and L'Hôpital's Rule since we have a $\frac{0}{0}$ form. $\newline$ $\lim_{x \to -1} \frac{1 + \sqrt{5x + 30}}{(x - 1)(x + 1)}$ $\newline$ Differentiate the numerator and the denominator separately. $\newline$ Numerator: $\newline$ $\frac{d}{dx} (1 + \sqrt{5x + 30}) = \frac{d}{dx} (1) + \frac{d}{dx} (\sqrt{5x + 30}) = 0 + \frac{5}{2\sqrt{5x + 30}} = \frac{5}{2\sqrt{5x + 30}}$ $\newline$ Denominator: $\newline$ $\frac{d}{dx} ((x - 1)(x + 1)) = \frac{d}{dx} (x^2 - 1) = 2x$ $\newline$ So, $\newline$ $\lim_{x \to -1} \frac{\frac{5}{2\sqrt{5x + 30}}}{2x} = \lim_{x \to -1} \frac{5}{4x\sqrt{5x + 30}}$
Simplify Limit: Substitute $x = -1$ into the simplified limit. $\newline$ $\lim_{x \to -1} \frac{5}{4x\sqrt{5x + 30}} = \frac{5}{4(-1)\sqrt{5(-1) + 30}} = \frac{5}{-4\sqrt{25}} = \frac{5}{-4 \cdot 5} = \frac{5}{-20} = -\frac{1}{4}$

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