Calculus Examples

$lim_{x \to 4} \frac{x - 2}{x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} x - 2}{lim_{x \to 4} x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} x - lim_{x \to 4} 2}{lim_{x \to 4} x}$

Step 3

Evaluate the limit of $2$ which is constant as $x$ approaches $4$ .

$\frac{lim_{x \to 4} x - 1 \cdot 2}{lim_{x \to 4} x}$

Step 4

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{4 - 1 \cdot 2}{lim_{x \to 4} x}$

Step 4.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{4 - 1 \cdot 2}{4}$

Step 5

Simplify the answer.

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Step 5.1

Cancel the common factor of $4 - 1 \cdot 2$ and $4$ .

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Step 5.1.1

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{- 1 (- 4) - 1 \cdot 2}{4}$

Step 5.1.2

Factor $- 1$ out of $- 1 \cdot 2$ .

$\frac{- 1 (- 4) - 1 (2)}{4}$

Step 5.1.3

Factor $- 1$ out of $- 1 (- 4) - 1 (2)$ .

$\frac{- 1 (- 4 + 2)}{4}$

Step 5.1.4

Factor $2$ out of $- 1 (- 4 + 2)$ .

$\frac{2 (- 1 (- 2 + 1))}{4}$

Step 5.1.5

Cancel the common factors.

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Step 5.1.5.1

Factor $2$ out of $4$ .

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

Step 5.1.5.2

Cancel the common factor.

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

Step 5.1.5.3

Rewrite the expression.

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

Step 5.2

Add $- 2$ and $1$ .

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

Step 5.3

Multiply $- 1$ by $- 1$ .

$\frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$