Calculus Examples

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

Step 1

Combine terms.

Tap for more steps...

Step 1.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Simplify the limit argument.

Tap for more steps...

Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.2

Multiply $\frac{x - 4}{x}$ by $\frac{1}{4 - x}$ .

$lim_{x \to 4} \frac{x - 4}{x (4 - x)}$

Step 2.1.3

Cancel the common factor of $x - 4$ and $4 - x$ .

Tap for more steps...

Step 2.1.3.1

Factor $- 1$ out of $x$ .

$lim_{x \to 4} \frac{- 1 (- x) - 4}{x (4 - x)}$

Step 2.1.3.2

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

$lim_{x \to 4} \frac{- 1 (- x) - 1 (4)}{x (4 - x)}$

Step 2.1.3.3

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

$lim_{x \to 4} \frac{- 1 (- x + 4)}{x (4 - x)}$

Step 2.1.3.4

Reorder terms.

$lim_{x \to 4} \frac{- 1 (- x + 4)}{x (- x + 4)}$

Step 2.1.3.5

Cancel the common factor.

$lim_{x \to 4} \frac{- 1 (- x + 4)}{x (- x + 4)}$

Step 2.1.3.6

Rewrite the expression.

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

Step 2.2

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

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

Step 2.3

Evaluate the limit of $- 1$ which is constant as $x$ approaches $4$ .

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

Step 3

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

$\frac{- 1}{4}$

Step 4

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$