Calculus Examples

$lim_{x \to 1} - \frac{x - 4}{x^{2} - 6 x + 8}$

Step 1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to 1} \frac{x - 4}{x^{2} - 6 x + 8}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$- \frac{lim_{x \to 1} x - 4}{lim_{x \to 1} x^{2} - 6 x + 8}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$- \frac{lim_{x \to 1} x - lim_{x \to 1} 4}{lim_{x \to 1} x^{2} - 6 x + 8}$

Step 4

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

$- \frac{lim_{x \to 1} x - 1 \cdot 4}{lim_{x \to 1} x^{2} - 6 x + 8}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$- \frac{lim_{x \to 1} x - 1 \cdot 4}{lim_{x \to 1} x^{2} - lim_{x \to 1} 6 x + lim_{x \to 1} 8}$

Step 6

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$- \frac{lim_{x \to 1} x - 1 \cdot 4}{{(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 6 x + lim_{x \to 1} 8}$

Step 7

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$- \frac{lim_{x \to 1} x - 1 \cdot 4}{{(lim_{x \to 1} x)}^{2} - 6 lim_{x \to 1} x + lim_{x \to 1} 8}$

Step 8

Evaluate the limit of $8$ which is constant as $x$ approaches $1$ .

$- \frac{lim_{x \to 1} x - 1 \cdot 4}{{(lim_{x \to 1} x)}^{2} - 6 lim_{x \to 1} x + 8}$

Step 9

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$- \frac{1 - 1 \cdot 4}{{(lim_{x \to 1} x)}^{2} - 6 lim_{x \to 1} x + 8}$

Step 9.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$- \frac{1 - 1 \cdot 4}{1^{2} - 6 lim_{x \to 1} x + 8}$

Step 9.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$- \frac{1 - 1 \cdot 4}{1^{2} - 6 \cdot 1 + 8}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $- 1$ by $4$ .

$- \frac{1 - 4}{1^{2} - 6 \cdot 1 + 8}$

Step 10.1.2

Subtract $4$ from $1$ .

$- \frac{- 3}{1^{2} - 6 \cdot 1 + 8}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

One to any power is one.

$- \frac{- 3}{1 - 6 \cdot 1 + 8}$

Step 10.2.2

Multiply $- 6$ by $1$ .

$- \frac{- 3}{1 - 6 + 8}$

Step 10.2.3

Subtract $6$ from $1$ .

$- \frac{- 3}{- 5 + 8}$

Step 10.2.4

Add $- 5$ and $8$ .

$- \frac{- 3}{3}$

Step 10.3

Divide $- 3$ by $3$ .

$- - 1$

Step 10.4

Multiply $- 1$ by $- 1$ .

$1$