Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Evaluate the limit of $4$ which is constant as $x$ approaches $- 1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Evaluate the limit of $3$ which is constant as $x$ approaches $- 1$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

$\frac{{(- 1)}^{2} - 3 \cdot - 1 - 1 \cdot 4}{{(- 1)}^{2} + 2 \cdot - 1 - 1 \cdot 3}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $- 1$ to the power of $2$ .

$\frac{1 - 3 \cdot - 1 - 1 \cdot 4}{{(- 1)}^{2} + 2 \cdot - 1 - 1 \cdot 3}$

Step 11.1.2

Multiply $- 3$ by $- 1$ .

$\frac{1 + 3 - 1 \cdot 4}{{(- 1)}^{2} + 2 \cdot - 1 - 1 \cdot 3}$

Step 11.1.3

Multiply $- 1$ by $4$ .

$\frac{1 + 3 - 4}{{(- 1)}^{2} + 2 \cdot - 1 - 1 \cdot 3}$

Step 11.1.4

Add $1$ and $3$ .

$\frac{4 - 4}{{(- 1)}^{2} + 2 \cdot - 1 - 1 \cdot 3}$

Step 11.1.5

Subtract $4$ from $4$ .

$\frac{0}{{(- 1)}^{2} + 2 \cdot - 1 - 1 \cdot 3}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Raise $- 1$ to the power of $2$ .

$\frac{0}{1 + 2 \cdot - 1 - 1 \cdot 3}$

Step 11.2.2

Multiply $2$ by $- 1$ .

$\frac{0}{1 - 2 - 1 \cdot 3}$

Step 11.2.3

Multiply $- 1$ by $3$ .

$\frac{0}{1 - 2 - 3}$

Step 11.2.4

Subtract $2$ from $1$ .

$\frac{0}{- 1 - 3}$

Step 11.2.5

Subtract $3$ from $- 1$ .

$\frac{0}{- 4}$

Step 11.3

Divide $0$ by $- 4$ .

$0$