Calculus Examples

$lim_{x \to - 5} \frac{4 f (x) + x - 2}{f (x) g (x) + \sqrt{f (x)}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 5$ .

$\frac{lim_{x \to - 5} 4 f (x) + x - 2}{lim_{x \to - 5} f (x) g (x) + \sqrt{f (x)}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 5$ .

$\frac{lim_{x \to - 5} 4 f (x) + lim_{x \to - 5} x - lim_{x \to - 5} 2}{lim_{x \to - 5} f (x) g (x) + \sqrt{f (x)}}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to - 5} f (x) + lim_{x \to - 5} x - lim_{x \to - 5} 2}{lim_{x \to - 5} f (x) g (x) + \sqrt{f (x)}}$

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $- 5$ .

$\frac{4 lim_{x \to - 5} f (x) + lim_{x \to - 5} x - 1 \cdot 2}{lim_{x \to - 5} f (x) g (x) + \sqrt{f (x)}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 5$ .

$\frac{4 lim_{x \to - 5} f (x) + lim_{x \to - 5} x - 1 \cdot 2}{lim_{x \to - 5} f (x) g (x) + lim_{x \to - 5} \sqrt{f (x)}}$

Step 6

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 5$ .

$\frac{4 lim_{x \to - 5} f (x) + lim_{x \to - 5} x - 1 \cdot 2}{lim_{x \to - 5} f (x) \cdot lim_{x \to - 5} g (x) + lim_{x \to - 5} \sqrt{f (x)}}$

Step 7

Move the limit under the radical sign.

$\frac{4 lim_{x \to - 5} f (x) + lim_{x \to - 5} x - 1 \cdot 2}{lim_{x \to - 5} f (x) \cdot lim_{x \to - 5} g (x) + \sqrt{lim_{x \to - 5} f (x)}}$

Step 8

Evaluate the limits by plugging in $- 5$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $f (x)$ by plugging in $- 5$ for $x$ .

$\frac{4 f (- 5) + lim_{x \to - 5} x - 1 \cdot 2}{lim_{x \to - 5} f (x) \cdot lim_{x \to - 5} g (x) + \sqrt{lim_{x \to - 5} f (x)}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $- 5$ for $x$ .

$\frac{4 f (- 5) - 5 - 1 \cdot 2}{lim_{x \to - 5} f (x) \cdot lim_{x \to - 5} g (x) + \sqrt{lim_{x \to - 5} f (x)}}$

Step 8.3

Evaluate the limit of $f (x)$ by plugging in $- 5$ for $x$ .

$\frac{4 f (- 5) - 5 - 1 \cdot 2}{f (- 5) \cdot lim_{x \to - 5} g (x) + \sqrt{lim_{x \to - 5} f (x)}}$

Step 8.4

Evaluate the limit of $g (x)$ by plugging in $- 5$ for $x$ .

$\frac{4 f (- 5) - 5 - 1 \cdot 2}{f (- 5) \cdot g (- 5) + \sqrt{lim_{x \to - 5} f (x)}}$

Step 8.5

Evaluate the limit of $f (x)$ by plugging in $- 5$ for $x$ .

$\frac{4 f (- 5) - 5 - 1 \cdot 2}{f (- 5) \cdot g (- 5) + \sqrt{f (- 5)}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $- 1$ by $2$ .

$\frac{4 f (- 5) - 5 - 2}{f (- 5) \cdot g (- 5) + \sqrt{f (- 5)}}$

Step 9.1.2

Subtract $2$ from $- 5$ .

$\frac{4 f (- 5) - 7}{f (- 5) \cdot g (- 5) + \sqrt{f (- 5)}}$

Step 9.2

Multiply $\frac{4 f (- 5) - 7}{f (- 5) \cdot g (- 5) + \sqrt{f (- 5)}}$ by $\frac{f (- 5) \cdot g (- 5) - \sqrt{f (- 5)}}{f (- 5) \cdot g (- 5) - \sqrt{f (- 5)}}$ .

$\frac{4 f (- 5) - 7}{f (- 5) \cdot g (- 5) + \sqrt{f (- 5)}} \cdot \frac{f (- 5) \cdot g (- 5) - \sqrt{f (- 5)}}{f (- 5) \cdot g (- 5) - \sqrt{f (- 5)}}$

Step 9.3

Multiply $\frac{4 f (- 5) - 7}{f (- 5) \cdot g (- 5) + \sqrt{f (- 5)}}$ by $\frac{f (- 5) \cdot g (- 5) - \sqrt{f (- 5)}}{f (- 5) \cdot g (- 5) - \sqrt{f (- 5)}}$ .

$\frac{(4 f (- 5) - 7) (f (- 5) \cdot g (- 5) - \sqrt{f (- 5)})}{(f (- 5) \cdot g (- 5) + \sqrt{f (- 5)}) (f (- 5) \cdot g (- 5) - \sqrt{f (- 5)})}$

Step 9.4

Expand the denominator using the FOIL method.

$\frac{(4 f (- 5) - 7) (f (- 5) \cdot g (- 5) - \sqrt{f (- 5)})}{{f (- 5)}^{2} {g (- 5)}^{2} + f (- 5) g (- 5) (- \sqrt{f (- 5)}) + \sqrt{f (- 5)} (f (- 5) g (- 5)) - {\sqrt{f (- 5)}}^{2}}$

Step 9.5

Simplify.

$\frac{(4 f (- 5) - 7) (f (- 5) g (- 5) - \sqrt{f (- 5)})}{{f (- 5)}^{2} {g (- 5)}^{2} - f (- 5)}$