Precalculus Examples

$lim_{x \to 3} \frac{5 x^{2} - 8 x - 13}{x^{2} - 5}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} 5 x^{2} - 8 x - 13}{lim_{x \to 3} x^{2} - 5}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} 5 x^{2} - lim_{x \to 3} 8 x - lim_{x \to 3} 13}{lim_{x \to 3} x^{2} - 5}$

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to 3} x^{2} - lim_{x \to 3} 8 x - lim_{x \to 3} 13}{lim_{x \to 3} x^{2} - 5}$

Step 4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{5 {(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 8 x - lim_{x \to 3} 13}{lim_{x \to 3} x^{2} - 5}$

Step 5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 {(lim_{x \to 3} x)}^{2} - 8 lim_{x \to 3} x - lim_{x \to 3} 13}{lim_{x \to 3} x^{2} - 5}$

Step 6

Evaluate the limit of $13$ which is constant as $x$ approaches $3$ .

$\frac{5 {(lim_{x \to 3} x)}^{2} - 8 lim_{x \to 3} x - 1 \cdot 13}{lim_{x \to 3} x^{2} - 5}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{5 {(lim_{x \to 3} x)}^{2} - 8 lim_{x \to 3} x - 1 \cdot 13}{lim_{x \to 3} x^{2} - lim_{x \to 3} 5}$

Step 8

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{5 {(lim_{x \to 3} x)}^{2} - 8 lim_{x \to 3} x - 1 \cdot 13}{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 5}$

Step 9

Evaluate the limit of $5$ which is constant as $x$ approaches $3$ .

$\frac{5 {(lim_{x \to 3} x)}^{2} - 8 lim_{x \to 3} x - 1 \cdot 13}{{(lim_{x \to 3} x)}^{2} - 1 \cdot 5}$

Step 10

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{5 \cdot 3^{2} - 8 lim_{x \to 3} x - 1 \cdot 13}{{(lim_{x \to 3} x)}^{2} - 1 \cdot 5}$

Step 10.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{5 \cdot 3^{2} - 8 \cdot 3 - 1 \cdot 13}{{(lim_{x \to 3} x)}^{2} - 1 \cdot 5}$

Step 10.3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{5 \cdot 3^{2} - 8 \cdot 3 - 1 \cdot 13}{3^{2} - 1 \cdot 5}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $3$ to the power of $2$ .

$\frac{5 \cdot 9 - 8 \cdot 3 - 1 \cdot 13}{3^{2} - 1 \cdot 5}$

Step 11.1.2

Multiply $5$ by $9$ .

$\frac{45 - 8 \cdot 3 - 1 \cdot 13}{3^{2} - 1 \cdot 5}$

Step 11.1.3

Multiply $- 8$ by $3$ .

$\frac{45 - 24 - 1 \cdot 13}{3^{2} - 1 \cdot 5}$

Step 11.1.4

Multiply $- 1$ by $13$ .

$\frac{45 - 24 - 13}{3^{2} - 1 \cdot 5}$

Step 11.1.5

Subtract $24$ from $45$ .

$\frac{21 - 13}{3^{2} - 1 \cdot 5}$

Step 11.1.6

Subtract $13$ from $21$ .

$\frac{8}{3^{2} - 1 \cdot 5}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Raise $3$ to the power of $2$ .

$\frac{8}{9 - 1 \cdot 5}$

Step 11.2.2

Multiply $- 1$ by $5$ .

$\frac{8}{9 - 5}$

Step 11.2.3

Subtract $5$ from $9$ .

$\frac{8}{4}$

Step 11.3

Divide $8$ by $4$ .

$2$