Calculus Examples

$lim_{x \to 4.001} \frac{4 - x}{| 4 - x |}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4.001$ .

$\frac{lim_{x \to 4.001} 4 - x}{lim_{x \to 4.001} | 4 - x |}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4.001$ .

$\frac{lim_{x \to 4.001} 4 - lim_{x \to 4.001} x}{lim_{x \to 4.001} | 4 - x |}$

Step 3

Evaluate the limit of $4$ which is constant as $x$ approaches $4.001$ .

$\frac{4 - lim_{x \to 4.001} x}{lim_{x \to 4.001} | 4 - x |}$

Step 4

Move the limit inside the absolute value signs.

$\frac{4 - lim_{x \to 4.001} x}{| lim_{x \to 4.001} 4 - x |}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4.001$ .

$\frac{4 - lim_{x \to 4.001} x}{| lim_{x \to 4.001} 4 - lim_{x \to 4.001} x |}$

Step 6

Evaluate the limit of $4$ which is constant as $x$ approaches $4.001$ .

$\frac{4 - lim_{x \to 4.001} x}{| 4 - lim_{x \to 4.001} x |}$

Step 7

Evaluate the limits by plugging in $4.001$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $4.001$ for $x$ .

$\frac{4 - 4.001}{| 4 - lim_{x \to 4.001} x |}$

Step 7.2

Evaluate the limit of $x$ by plugging in $4.001$ for $x$ .

$\frac{4 - 4.001}{| 4 - 4.001 |}$

Step 8

Simplify the answer.

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Step 8.1

Subtract $4.001$ from $4$ .

$\frac{- 0.001}{| 4 - 4.001 |}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Subtract $4.001$ from $4$ .

$\frac{- 0.001}{| - 0.001 |}$

Step 8.2.2

The absolute value is the distance between a number and zero. The distance between $- 0.001$ and $0$ is $0.001$ .

$\frac{- 0.001}{0.001}$

Step 8.3

Divide $- 0.001$ by $0.001$ .

$- 1$