Calculus Examples

$lim_{x \to - 5} \frac{x^{2} + 14 x + 45}{- 35 - 4 x + x^{2}}$

Step 1

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

$\frac{lim_{x \to - 5} x^{2} + 14 x + 45}{lim_{x \to - 5} - 35 - 4 x + x^{2}}$

Step 2

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

$\frac{lim_{x \to - 5} x^{2} + lim_{x \to - 5} 14 x + lim_{x \to - 5} 45}{lim_{x \to - 5} - 35 - 4 x + x^{2}}$

Step 3

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

$\frac{{(lim_{x \to - 5} x)}^{2} + lim_{x \to - 5} 14 x + lim_{x \to - 5} 45}{lim_{x \to - 5} - 35 - 4 x + x^{2}}$

Step 4

Move the term $14$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to - 5} x)}^{2} + 14 lim_{x \to - 5} x + lim_{x \to - 5} 45}{lim_{x \to - 5} - 35 - 4 x + x^{2}}$

Step 5

Evaluate the limit of $45$ which is constant as $x$ approaches $- 5$ .

$\frac{{(lim_{x \to - 5} x)}^{2} + 14 lim_{x \to - 5} x + 45}{lim_{x \to - 5} - 35 - 4 x + x^{2}}$

Step 6

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

$\frac{{(lim_{x \to - 5} x)}^{2} + 14 lim_{x \to - 5} x + 45}{- lim_{x \to - 5} 35 - lim_{x \to - 5} 4 x + lim_{x \to - 5} x^{2}}$

Step 7

Evaluate the limit of $35$ which is constant as $x$ approaches $- 5$ .

$\frac{{(lim_{x \to - 5} x)}^{2} + 14 lim_{x \to - 5} x + 45}{- 1 \cdot 35 - lim_{x \to - 5} 4 x + lim_{x \to - 5} x^{2}}$

Step 8

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

$\frac{{(lim_{x \to - 5} x)}^{2} + 14 lim_{x \to - 5} x + 45}{- 35 - 4 lim_{x \to - 5} x + lim_{x \to - 5} x^{2}}$

Step 9

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

$\frac{{(lim_{x \to - 5} x)}^{2} + 14 lim_{x \to - 5} x + 45}{- 35 - 4 lim_{x \to - 5} x + {(lim_{x \to - 5} x)}^{2}}$

Step 10

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

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Step 10.1

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

$\frac{{(- 5)}^{2} + 14 lim_{x \to - 5} x + 45}{- 35 - 4 lim_{x \to - 5} x + {(lim_{x \to - 5} x)}^{2}}$

Step 10.2

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

$\frac{{(- 5)}^{2} + 14 \cdot - 5 + 45}{- 35 - 4 lim_{x \to - 5} x + {(lim_{x \to - 5} x)}^{2}}$

Step 10.3

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

$\frac{{(- 5)}^{2} + 14 \cdot - 5 + 45}{- 35 - 4 \cdot - 5 + {(lim_{x \to - 5} x)}^{2}}$

Step 10.4

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

$\frac{{(- 5)}^{2} + 14 \cdot - 5 + 45}{- 35 - 4 \cdot - 5 + {(- 5)}^{2}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $- 5$ to the power of $2$ .

$\frac{25 + 14 \cdot - 5 + 45}{- 35 - 4 \cdot - 5 + {(- 5)}^{2}}$

Step 11.1.2

Multiply $14$ by $- 5$ .

$\frac{25 - 70 + 45}{- 35 - 4 \cdot - 5 + {(- 5)}^{2}}$

Step 11.1.3

Subtract $70$ from $25$ .

$\frac{- 45 + 45}{- 35 - 4 \cdot - 5 + {(- 5)}^{2}}$

Step 11.1.4

Add $- 45$ and $45$ .

$\frac{0}{- 35 - 4 \cdot - 5 + {(- 5)}^{2}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $- 4$ by $- 5$ .

$\frac{0}{- 35 + 20 + {(- 5)}^{2}}$

Step 11.2.2

Raise $- 5$ to the power of $2$ .

$\frac{0}{- 35 + 20 + 25}$

Step 11.2.3

Add $- 35$ and $20$ .

$\frac{0}{- 15 + 25}$

Step 11.2.4

Add $- 15$ and $25$ .

$\frac{0}{10}$

Step 11.3

Divide $0$ by $10$ .

$0$