Calculus Examples

$\frac{lim_{x \to - 9} x^{2} - 81}{2 x^{2} + 19 x + 9}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 9$ .

$\frac{lim_{x \to - 9} x^{2} - lim_{x \to - 9} 81}{2 x^{2} + 19 x + 9}$

Step 1.2

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

$\frac{{(lim_{x \to - 9} x)}^{2} - lim_{x \to - 9} 81}{2 x^{2} + 19 x + 9}$

Step 1.3

Evaluate the limit of $81$ which is constant as $x$ approaches $- 9$ .

$\frac{{(lim_{x \to - 9} x)}^{2} - 1 \cdot 81}{2 x^{2} + 19 x + 9}$

Step 2

Evaluate the limit of $x$ by plugging in $- 9$ for $x$ .

$\frac{{(- 9)}^{2} - 1 \cdot 81}{2 x^{2} + 19 x + 9}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 9$ to the power of $2$ .

$\frac{81 - 1 \cdot 81}{2 x^{2} + 19 x + 9}$

Step 3.1.2

Multiply $- 1$ by $81$ .

$\frac{81 - 81}{2 x^{2} + 19 x + 9}$

Step 3.1.3

Subtract $81$ from $81$ .

$\frac{0}{2 x^{2} + 19 x + 9}$

Step 3.2

Factor by grouping.

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Step 3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 9 = 18$ and whose sum is $b = 19$ .

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Step 3.2.1.1

Factor $19$ out of $19 x$ .

$\frac{0}{2 x^{2} + 19 (x) + 9}$

Step 3.2.1.2

Rewrite $19$ as $1$ plus $18$

$\frac{0}{2 x^{2} + (1 + 18) x + 9}$

Step 3.2.1.3

Apply the distributive property.

$\frac{0}{2 x^{2} + 1 x + 18 x + 9}$

Step 3.2.1.4

Multiply $x$ by $1$ .

$\frac{0}{2 x^{2} + x + 18 x + 9}$

Step 3.2.2

Factor out the greatest common factor from each group.

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Step 3.2.2.1

Group the first two terms and the last two terms.

$\frac{0}{(2 x^{2} + x) + 18 x + 9}$

Step 3.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{0}{x (2 x + 1) + 9 (2 x + 1)}$

Step 3.2.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$\frac{0}{(2 x + 1) (x + 9)}$

Step 3.3

Divide $0$ by $(2 x + 1) (x + 9)$ .

$0$