Calculus Examples

$\frac{lim_{x \to 1} x^{2} + 7 x - 8}{x^{2} - 1}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} x^{2} + lim_{x \to 1} 7 x - lim_{x \to 1} 8}{x^{2} - 1}$

Step 2

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

$\frac{{(lim_{x \to 1} x)}^{2} + lim_{x \to 1} 7 x - lim_{x \to 1} 8}{x^{2} - 1}$

Step 3

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

$\frac{{(lim_{x \to 1} x)}^{2} + 7 lim_{x \to 1} x - lim_{x \to 1} 8}{x^{2} - 1}$

Step 4

Evaluate the limit of $8$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{2} + 7 lim_{x \to 1} x - 1 \cdot 8}{x^{2} - 1}$

Step 5

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{2} + 7 lim_{x \to 1} x - 1 \cdot 8}{x^{2} - 1}$

Step 5.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{2} + 7 \cdot 1 - 1 \cdot 8}{x^{2} - 1}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

One to any power is one.

$\frac{1 + 7 \cdot 1 - 1 \cdot 8}{x^{2} - 1}$

Step 6.1.2

Multiply $7$ by $1$ .

$\frac{1 + 7 - 1 \cdot 8}{x^{2} - 1}$

Step 6.1.3

Multiply $- 1$ by $8$ .

$\frac{1 + 7 - 8}{x^{2} - 1}$

Step 6.1.4

Add $1$ and $7$ .

$\frac{8 - 8}{x^{2} - 1}$

Step 6.1.5

Subtract $8$ from $8$ .

$\frac{0}{x^{2} - 1}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{0}{x^{2} - 1^{2}}$

Step 6.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{0}{(x + 1) (x - 1)}$

Step 6.3

Divide $0$ by $(x + 1) (x - 1)$ .

$0$