Calculus Examples

$lim_{x \to 1} \frac{x^{2} + 4 x - 5}{| x^{2} - 4 |}$

Step 1

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

$\frac{lim_{x \to 1} x^{2} + 4 x - 5}{lim_{x \to 1} | x^{2} - 4 |}$

Step 2

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

$\frac{lim_{x \to 1} x^{2} + lim_{x \to 1} 4 x - lim_{x \to 1} 5}{lim_{x \to 1} | x^{2} - 4 |}$

Step 3

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

$\frac{{(lim_{x \to 1} x)}^{2} + lim_{x \to 1} 4 x - lim_{x \to 1} 5}{lim_{x \to 1} | x^{2} - 4 |}$

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 1} x)}^{2} + 4 lim_{x \to 1} x - lim_{x \to 1} 5}{lim_{x \to 1} | x^{2} - 4 |}$

Step 5

Evaluate the limit of $5$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{2} + 4 lim_{x \to 1} x - 1 \cdot 5}{lim_{x \to 1} | x^{2} - 4 |}$

Step 6

Move the limit inside the absolute value signs.

$\frac{{(lim_{x \to 1} x)}^{2} + 4 lim_{x \to 1} x - 1 \cdot 5}{| lim_{x \to 1} x^{2} - 4 |}$

Step 7

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

$\frac{{(lim_{x \to 1} x)}^{2} + 4 lim_{x \to 1} x - 1 \cdot 5}{| lim_{x \to 1} x^{2} - lim_{x \to 1} 4 |}$

Step 8

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

$\frac{{(lim_{x \to 1} x)}^{2} + 4 lim_{x \to 1} x - 1 \cdot 5}{| {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 4 |}$

Step 9

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

$\frac{{(lim_{x \to 1} x)}^{2} + 4 lim_{x \to 1} x - 1 \cdot 5}{| {(lim_{x \to 1} x)}^{2} - 1 \cdot 4 |}$

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 $x$ .

$\frac{1^{2} + 4 lim_{x \to 1} x - 1 \cdot 5}{| {(lim_{x \to 1} x)}^{2} - 1 \cdot 4 |}$

Step 10.2

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

$\frac{1^{2} + 4 \cdot 1 - 1 \cdot 5}{| {(lim_{x \to 1} x)}^{2} - 1 \cdot 4 |}$

Step 10.3

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

$\frac{1^{2} + 4 \cdot 1 - 1 \cdot 5}{| 1^{2} - 1 \cdot 4 |}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

One to any power is one.

$\frac{1 + 4 \cdot 1 - 1 \cdot 5}{| 1^{2} - 1 \cdot 4 |}$

Step 11.1.2

Multiply $4$ by $1$ .

$\frac{1 + 4 - 1 \cdot 5}{| 1^{2} - 1 \cdot 4 |}$

Step 11.1.3

Multiply $- 1$ by $5$ .

$\frac{1 + 4 - 5}{| 1^{2} - 1 \cdot 4 |}$

Step 11.1.4

Add $1$ and $4$ .

$\frac{5 - 5}{| 1^{2} - 1 \cdot 4 |}$

Step 11.1.5

Subtract $5$ from $5$ .

$\frac{0}{| 1^{2} - 1 \cdot 4 |}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

One to any power is one.

$\frac{0}{| 1 - 1 \cdot 4 |}$

Step 11.2.2

Multiply $- 1$ by $4$ .

$\frac{0}{| 1 - 4 |}$

Step 11.2.3

Subtract $4$ from $1$ .

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

Step 11.2.4

The absolute value is the distance between a number and zero. The distance between $- 3$ and $0$ is $3$ .

$\frac{0}{3}$

Step 11.3

Divide $0$ by $3$ .

$0$