Calculus Examples

$lim_{x \to 0} \frac{2 x^{2} + 7 x + 6}{{(x - 4)}^{2}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 2 x^{2} + 7 x + 6}{lim_{x \to 0} {(x - 4)}^{2}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 2 x^{2} + lim_{x \to 0} 7 x + lim_{x \to 0} 6}{lim_{x \to 0} {(x - 4)}^{2}}$

Step 3

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

$\frac{2 lim_{x \to 0} x^{2} + lim_{x \to 0} 7 x + lim_{x \to 0} 6}{lim_{x \to 0} {(x - 4)}^{2}}$

Step 4

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

$\frac{2 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 7 x + lim_{x \to 0} 6}{lim_{x \to 0} {(x - 4)}^{2}}$

Step 5

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 {(lim_{x \to 0} x)}^{2} + 7 lim_{x \to 0} x + lim_{x \to 0} 6}{lim_{x \to 0} {(x - 4)}^{2}}$

Step 6

Evaluate the limit of $6$ which is constant as $x$ approaches $0$ .

$\frac{2 {(lim_{x \to 0} x)}^{2} + 7 lim_{x \to 0} x + 6}{lim_{x \to 0} {(x - 4)}^{2}}$

Step 7

Move the exponent $2$ from ${(x - 4)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{2 {(lim_{x \to 0} x)}^{2} + 7 lim_{x \to 0} x + 6}{{(lim_{x \to 0} x - 4)}^{2}}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{2 {(lim_{x \to 0} x)}^{2} + 7 lim_{x \to 0} x + 6}{{(lim_{x \to 0} x - lim_{x \to 0} 4)}^{2}}$

Step 9

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$\frac{2 {(lim_{x \to 0} x)}^{2} + 7 lim_{x \to 0} x + 6}{{(lim_{x \to 0} x - 1 \cdot 4)}^{2}}$

Step 10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2 \cdot 0^{2} + 7 lim_{x \to 0} x + 6}{{(lim_{x \to 0} x - 1 \cdot 4)}^{2}}$

Step 10.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2 \cdot 0^{2} + 7 \cdot 0 + 6}{{(lim_{x \to 0} x - 1 \cdot 4)}^{2}}$

Step 10.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2 \cdot 0^{2} + 7 \cdot 0 + 6}{{(0 - 1 \cdot 4)}^{2}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{2 \cdot 0 + 7 \cdot 0 + 6}{{(0 - 1 \cdot 4)}^{2}}$

Step 11.1.2

Multiply $2$ by $0$ .

$\frac{0 + 7 \cdot 0 + 6}{{(0 - 1 \cdot 4)}^{2}}$

Step 11.1.3

Multiply $7$ by $0$ .

$\frac{0 + 0 + 6}{{(0 - 1 \cdot 4)}^{2}}$

Step 11.1.4

Add $0$ and $0$ .

$\frac{0 + 6}{{(0 - 1 \cdot 4)}^{2}}$

Step 11.1.5

Add $0$ and $6$ .

$\frac{6}{{(0 - 1 \cdot 4)}^{2}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $- 1$ by $4$ .

$\frac{6}{{(0 - 4)}^{2}}$

Step 11.2.2

Subtract $4$ from $0$ .

$\frac{6}{{(- 4)}^{2}}$

Step 11.2.3

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

$\frac{6}{16}$

Step 11.3

Cancel the common factor of $6$ and $16$ .

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Step 11.3.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{16}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $2$ out of $16$ .

$\frac{2 \cdot 3}{2 \cdot 8}$

Step 11.3.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 \cdot 8}$

Step 11.3.2.3

Rewrite the expression.

$\frac{3}{8}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{8}$

Decimal Form:

$0.375$