Calculus Examples

$lim_{x \to 3} 2 x + | x - 3 |$

Step 1

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

$lim_{x \to 3} 2 x + lim_{x \to 3} | x - 3 |$

Step 2

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

$2 lim_{x \to 3} x + lim_{x \to 3} | x - 3 |$

Step 3

Move the limit inside the absolute value signs.

$2 lim_{x \to 3} x + | lim_{x \to 3} x - 3 |$

Step 4

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

$2 lim_{x \to 3} x + | lim_{x \to 3} x - lim_{x \to 3} 3 |$

Step 5

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

$2 lim_{x \to 3} x + | lim_{x \to 3} x - 1 \cdot 3 |$

Step 6

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

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Step 6.1

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

$2 \cdot 3 + | lim_{x \to 3} x - 1 \cdot 3 |$

Step 6.2

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

$2 \cdot 3 + | 3 - 1 \cdot 3 |$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $2$ by $3$ .

$6 + | 3 - 1 \cdot 3 |$

Step 7.1.2

Multiply $- 1$ by $3$ .

$6 + | 3 - 3 |$

Step 7.1.3

Subtract $3$ from $3$ .

$6 + | 0 |$

Step 7.1.4

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

$6 + 0$

Step 7.2

Add $6$ and $0$ .

$6$