Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 3

Evaluate the limit of $2$ which is constant as $x$ approaches $2$ .

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

Step 4

Move the limit inside the absolute value signs.

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

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 6

Evaluate the limit of $2$ which is constant as $x$ approaches $2$ .

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

Step 7

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

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

Step 7.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2 + | 2 |}{2 + 2}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$\frac{2 + 2}{2 + 2}$

Step 8.1.2

Add $2$ and $2$ .

$\frac{4}{2 + 2}$

Step 8.2

Add $2$ and $2$ .

$\frac{4}{4}$

Step 8.3

Divide $4$ by $4$ .

$1$