Precalculus Examples

$\frac{lim_{x \to 4} (3 x^{2} + 2 x) - 56}{2 x^{2} + 3 x - 44}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} 3 x^{2} + lim_{x \to 4} 2 x - lim_{x \to 4} 56}{2 x^{2} + 3 x - 44}$

Step 2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 4} x^{2} + lim_{x \to 4} 2 x - lim_{x \to 4} 56}{2 x^{2} + 3 x - 44}$

Step 3

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

$\frac{3 {(lim_{x \to 4} x)}^{2} + lim_{x \to 4} 2 x - lim_{x \to 4} 56}{2 x^{2} + 3 x - 44}$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to 4} x)}^{2} + 2 lim_{x \to 4} x - lim_{x \to 4} 56}{2 x^{2} + 3 x - 44}$

Step 5

Evaluate the limit of $56$ which is constant as $x$ approaches $4$ .

$\frac{3 {(lim_{x \to 4} x)}^{2} + 2 lim_{x \to 4} x - 1 \cdot 56}{2 x^{2} + 3 x - 44}$

Step 6

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{3 \cdot 4^{2} + 2 lim_{x \to 4} x - 1 \cdot 56}{2 x^{2} + 3 x - 44}$

Step 6.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{3 \cdot 4^{2} + 2 \cdot 4 - 1 \cdot 56}{2 x^{2} + 3 x - 44}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $4$ to the power of $2$ .

$\frac{3 \cdot 16 + 2 \cdot 4 - 1 \cdot 56}{2 x^{2} + 3 x - 44}$

Step 7.1.2

Multiply $3$ by $16$ .

$\frac{48 + 2 \cdot 4 - 1 \cdot 56}{2 x^{2} + 3 x - 44}$

Step 7.1.3

Multiply $2$ by $4$ .

$\frac{48 + 8 - 1 \cdot 56}{2 x^{2} + 3 x - 44}$

Step 7.1.4

Multiply $- 1$ by $56$ .

$\frac{48 + 8 - 56}{2 x^{2} + 3 x - 44}$

Step 7.1.5

Add $48$ and $8$ .

$\frac{56 - 56}{2 x^{2} + 3 x - 44}$

Step 7.1.6

Subtract $56$ from $56$ .

$\frac{0}{2 x^{2} + 3 x - 44}$

Step 7.2

Factor by grouping.

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Step 7.2.1

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 $a \cdot c = 2 \cdot - 44 = - 88$ and whose sum is $b = 3$ .

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Step 7.2.1.1

Factor $3$ out of $3 x$ .

$\frac{0}{2 x^{2} + 3 (x) - 44}$

Step 7.2.1.2

Rewrite $3$ as $- 8$ plus $11$

$\frac{0}{2 x^{2} + (- 8 + 11) x - 44}$

Step 7.2.1.3

Apply the distributive property.

$\frac{0}{2 x^{2} - 8 x + 11 x - 44}$

Step 7.2.2

Factor out the greatest common factor from each group.

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Step 7.2.2.1

Group the first two terms and the last two terms.

$\frac{0}{(2 x^{2} - 8 x) + 11 x - 44}$

Step 7.2.2.2

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

$\frac{0}{2 x (x - 4) + 11 (x - 4)}$

Step 7.2.3

Factor the polynomial by factoring out the greatest common factor, $x - 4$ .

$\frac{0}{(x - 4) (2 x + 11)}$

Step 7.3

Divide $0$ by $(x - 4) (2 x + 11)$ .

$0$