Calculus Examples

$lim_{x \to - 3} | x + 1 | + \frac{3}{x}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$lim_{x \to - 3} | x + 1 | + lim_{x \to - 3} \frac{3}{x}$

Step 2

Move the limit inside the absolute value signs.

$| lim_{x \to - 3} x + 1 | + lim_{x \to - 3} \frac{3}{x}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$| lim_{x \to - 3} x + lim_{x \to - 3} 1 | + lim_{x \to - 3} \frac{3}{x}$

Step 4

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

$| lim_{x \to - 3} x + 1 | + lim_{x \to - 3} \frac{3}{x}$

Step 5

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

$| lim_{x \to - 3} x + 1 | + 3 lim_{x \to - 3} \frac{1}{x}$

Step 6

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 3$ .

$| lim_{x \to - 3} x + 1 | + 3 \frac{lim_{x \to - 3} 1}{lim_{x \to - 3} x}$

Step 7

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

$| lim_{x \to - 3} x + 1 | + 3 \frac{1}{lim_{x \to - 3} x}$

Step 8

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$| - 3 + 1 | + 3 \frac{1}{lim_{x \to - 3} x}$

Step 8.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$| - 3 + 1 | + 3 (\frac{1}{- 3})$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Add $- 3$ and $1$ .

$| - 2 | + 3 (\frac{1}{- 3})$

Step 9.1.2

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

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

Step 9.1.3

Cancel the common factor of $3$ .

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Step 9.1.3.1

Factor $3$ out of $- 3$ .

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

Step 9.1.3.2

Cancel the common factor.

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

Step 9.1.3.3

Rewrite the expression.

$2 + \frac{1}{- 1}$

Step 9.1.4

Divide $1$ by $- 1$ .

$2 - 1$

Step 9.2

Subtract $1$ from $2$ .

$1$