Calculus Examples

$\frac{lim_{x \to - 2} 6 x^{2} + x - 2}{2 x^{2} + 3 x - 2}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{lim_{x \to - 2} 6 x^{2} + lim_{x \to - 2} x - lim_{x \to - 2} 2}{2 x^{2} + 3 x - 2}$

Step 2

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 lim_{x \to - 2} x^{2} + lim_{x \to - 2} x - lim_{x \to - 2} 2}{2 x^{2} + 3 x - 2}$

Step 3

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

$\frac{6 {(lim_{x \to - 2} x)}^{2} + lim_{x \to - 2} x - lim_{x \to - 2} 2}{2 x^{2} + 3 x - 2}$

Step 4

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

$\frac{6 {(lim_{x \to - 2} x)}^{2} + lim_{x \to - 2} x - 1 \cdot 2}{2 x^{2} + 3 x - 2}$

Step 5

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{6 {(- 2)}^{2} + lim_{x \to - 2} x - 1 \cdot 2}{2 x^{2} + 3 x - 2}$

Step 5.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{6 {(- 2)}^{2} - 2 - 1 \cdot 2}{2 x^{2} + 3 x - 2}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$\frac{6 \cdot 4 - 2 - 1 \cdot 2}{2 x^{2} + 3 x - 2}$

Step 6.1.2

Multiply $6$ by $4$ .

$\frac{24 - 2 - 1 \cdot 2}{2 x^{2} + 3 x - 2}$

Step 6.1.3

Multiply $- 1$ by $2$ .

$\frac{24 - 2 - 2}{2 x^{2} + 3 x - 2}$

Step 6.1.4

Subtract $2$ from $24$ .

$\frac{22 - 2}{2 x^{2} + 3 x - 2}$

Step 6.1.5

Subtract $2$ from $22$ .

$\frac{20}{2 x^{2} + 3 x - 2}$

Step 6.2

Factor by grouping.

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Step 6.2.1

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = 3$ .

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Step 6.2.1.1

Factor $3$ out of $3 x$ .

$\frac{20}{2 x^{2} + 3 (x) - 2}$

Step 6.2.1.2

Rewrite $3$ as $- 1$ plus $4$

$\frac{20}{2 x^{2} + (- 1 + 4) x - 2}$

Step 6.2.1.3

Apply the distributive property.

$\frac{20}{2 x^{2} - 1 x + 4 x - 2}$

Step 6.2.2

Factor out the greatest common factor from each group.

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Step 6.2.2.1

Group the first two terms and the last two terms.

$\frac{20}{(2 x^{2} - 1 x) + 4 x - 2}$

Step 6.2.2.2

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

$\frac{20}{x (2 x - 1) + 2 (2 x - 1)}$

Step 6.2.3

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

$\frac{20}{(2 x - 1) (x + 2)}$