Calculus Examples

$lim_{x \to 6} \frac{x^{2} + 5 x - 6}{x^{2} - 1}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $6$ .

$\frac{lim_{x \to 6} x^{2} + 5 x - 6}{lim_{x \to 6} x^{2} - 1}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{lim_{x \to 6} x^{2} + lim_{x \to 6} 5 x - lim_{x \to 6} 6}{lim_{x \to 6} x^{2} - 1}$

Step 3

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

$\frac{{(lim_{x \to 6} x)}^{2} + lim_{x \to 6} 5 x - lim_{x \to 6} 6}{lim_{x \to 6} x^{2} - 1}$

Step 4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 6} x)}^{2} + 5 lim_{x \to 6} x - lim_{x \to 6} 6}{lim_{x \to 6} x^{2} - 1}$

Step 5

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

$\frac{{(lim_{x \to 6} x)}^{2} + 5 lim_{x \to 6} x - 1 \cdot 6}{lim_{x \to 6} x^{2} - 1}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$\frac{{(lim_{x \to 6} x)}^{2} + 5 lim_{x \to 6} x - 1 \cdot 6}{lim_{x \to 6} x^{2} - lim_{x \to 6} 1}$

Step 7

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

$\frac{{(lim_{x \to 6} x)}^{2} + 5 lim_{x \to 6} x - 1 \cdot 6}{{(lim_{x \to 6} x)}^{2} - lim_{x \to 6} 1}$

Step 8

Evaluate the limit of $1$ which is constant as $x$ approaches $6$ .

$\frac{{(lim_{x \to 6} x)}^{2} + 5 lim_{x \to 6} x - 1 \cdot 6}{{(lim_{x \to 6} x)}^{2} - 1 \cdot 1}$

Step 9

Evaluate the limits by plugging in $6$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{2} + 5 lim_{x \to 6} x - 1 \cdot 6}{{(lim_{x \to 6} x)}^{2} - 1 \cdot 1}$

Step 9.2

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{2} + 5 \cdot 6 - 1 \cdot 6}{{(lim_{x \to 6} x)}^{2} - 1 \cdot 1}$

Step 9.3

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$\frac{6^{2} + 5 \cdot 6 - 1 \cdot 6}{6^{2} - 1 \cdot 1}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $6$ to the power of $2$ .

$\frac{36 + 5 \cdot 6 - 1 \cdot 6}{6^{2} - 1 \cdot 1}$

Step 10.1.2

Multiply $5$ by $6$ .

$\frac{36 + 30 - 1 \cdot 6}{6^{2} - 1 \cdot 1}$

Step 10.1.3

Multiply $- 1$ by $6$ .

$\frac{36 + 30 - 6}{6^{2} - 1 \cdot 1}$

Step 10.1.4

Add $36$ and $30$ .

$\frac{66 - 6}{6^{2} - 1 \cdot 1}$

Step 10.1.5

Subtract $6$ from $66$ .

$\frac{60}{6^{2} - 1 \cdot 1}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Raise $6$ to the power of $2$ .

$\frac{60}{36 - 1 \cdot 1}$

Step 10.2.2

Multiply $- 1$ by $1$ .

$\frac{60}{36 - 1}$

Step 10.2.3

Subtract $1$ from $36$ .

$\frac{60}{35}$

Step 10.3

Cancel the common factor of $60$ and $35$ .

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Step 10.3.1

Factor $5$ out of $60$ .

$\frac{5 (12)}{35}$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $5$ out of $35$ .

$\frac{5 \cdot 12}{5 \cdot 7}$

Step 10.3.2.2

Cancel the common factor.

$\frac{5 \cdot 12}{5 \cdot 7}$

Step 10.3.2.3

Rewrite the expression.

$\frac{12}{7}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{12}{7}$

Decimal Form:

$1.\overline{714285}$