Calculus Examples

$lim_{x \to 2} {(3 x - 8)}^{3} \cdot \frac{3 x}{5 x - 4}$

Step 1

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

$lim_{x \to 2} {(3 x - 8)}^{3} \cdot lim_{x \to 2} \frac{3 x}{5 x - 4}$

Step 2

Move the exponent $3$ from ${(3 x - 8)}^{3}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 2} 3 x - 8)}^{3} \cdot lim_{x \to 2} \frac{3 x}{5 x - 4}$

Step 3

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

${(lim_{x \to 2} 3 x - lim_{x \to 2} 8)}^{3} \cdot lim_{x \to 2} \frac{3 x}{5 x - 4}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

${(3 lim_{x \to 2} x - lim_{x \to 2} 8)}^{3} \cdot lim_{x \to 2} \frac{3 x}{5 x - 4}$

Step 5

Evaluate the limit of $8$ which is constant as $x$ approaches $2$ .

${(3 lim_{x \to 2} x - 1 \cdot 8)}^{3} \cdot lim_{x \to 2} \frac{3 x}{5 x - 4}$

Step 6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

${(3 lim_{x \to 2} x - 1 \cdot 8)}^{3} \cdot 3 lim_{x \to 2} \frac{x}{5 x - 4}$

Step 7

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

${(3 lim_{x \to 2} x - 1 \cdot 8)}^{3} \cdot 3 \frac{lim_{x \to 2} x}{lim_{x \to 2} 5 x - 4}$

Step 8

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

${(3 lim_{x \to 2} x - 1 \cdot 8)}^{3} \cdot 3 \frac{lim_{x \to 2} x}{lim_{x \to 2} 5 x - lim_{x \to 2} 4}$

Step 9

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

${(3 lim_{x \to 2} x - 1 \cdot 8)}^{3} \cdot 3 \frac{lim_{x \to 2} x}{5 lim_{x \to 2} x - lim_{x \to 2} 4}$

Step 10

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

${(3 lim_{x \to 2} x - 1 \cdot 8)}^{3} \cdot 3 \frac{lim_{x \to 2} x}{5 lim_{x \to 2} x - 1 \cdot 4}$

Step 11

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

Tap for more steps...

Step 11.1

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

${(3 \cdot 2 - 1 \cdot 8)}^{3} \cdot 3 \frac{lim_{x \to 2} x}{5 lim_{x \to 2} x - 1 \cdot 4}$

Step 11.2

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

${(3 \cdot 2 - 1 \cdot 8)}^{3} \cdot 3 \frac{2}{5 lim_{x \to 2} x - 1 \cdot 4}$

Step 11.3

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

${(3 \cdot 2 - 1 \cdot 8)}^{3} \cdot 3 \frac{2}{5 \cdot 2 - 1 \cdot 4}$

Step 12

Simplify the answer.

Tap for more steps...

Step 12.1

Simplify each term.

Tap for more steps...

Step 12.1.1

Multiply $3$ by $2$ .

${(6 - 1 \cdot 8)}^{3} \cdot 3 \frac{2}{5 \cdot 2 - 1 \cdot 4}$

Step 12.1.2

Multiply $- 1$ by $8$ .

${(6 - 8)}^{3} \cdot 3 \frac{2}{5 \cdot 2 - 1 \cdot 4}$

Step 12.2

Subtract $8$ from $6$ .

${(- 2)}^{3} \cdot 3 \frac{2}{5 \cdot 2 - 1 \cdot 4}$

Step 12.3

Raise $- 2$ to the power of $3$ .

$- 8 \cdot 3 \frac{2}{5 \cdot 2 - 1 \cdot 4}$

Step 12.4

Multiply $- 8$ by $3$ .

$- 24 \frac{2}{5 \cdot 2 - 1 \cdot 4}$

Step 12.5

Cancel the common factor of $2$ and $5 \cdot 2 - 1 \cdot 4$ .

Tap for more steps...

Step 12.5.1

Factor $- 1$ out of $5 \cdot 2$ .

$- 24 \frac{2}{- (- 5 \cdot 2) - 1 \cdot 4}$

Step 12.5.2

Factor $- 1$ out of $- 1 \cdot 4$ .

$- 24 \frac{2}{- (- 5 \cdot 2) - 1 (4)}$

Step 12.5.3

Factor $- 1$ out of $- (- 5 \cdot 2) - 1 (4)$ .

$- 24 \frac{2}{- (- 5 \cdot 2 + 4)}$

Step 12.5.4

Rewrite $- (- 5 \cdot 2 + 4)$ as $- 1 (- 5 \cdot 2 + 4)$ .

$- 24 \frac{2}{- 1 (- 5 \cdot 2 + 4)}$

Step 12.5.5

Factor $2$ out of $2$ .

$- 24 \frac{2 \cdot 1}{- 1 (- 5 \cdot 2 + 4)}$

Step 12.5.6

Cancel the common factors.

Tap for more steps...

Step 12.5.6.1

Factor $2$ out of $- 1 (- 5 \cdot 2 + 4)$ .

$- 24 \frac{2 \cdot 1}{2 (- 1 (- 5 + 2))}$

Step 12.5.6.2

Cancel the common factor.

$- 24 \frac{2 \cdot 1}{2 (- 1 (- 5 + 2))}$

Step 12.5.6.3

Rewrite the expression.

$- 24 \frac{1}{- 1 (- 5 + 2)}$

Step 12.6

Cancel the common factor of $1$ and $- 1$ .

Tap for more steps...

Step 12.6.1

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

$- 24 \frac{- 1 (- 1)}{- 1 (- 5 + 2)}$

Step 12.6.2

Move the negative in front of the fraction.

$- 24 (- \frac{1}{- 5 + 2})$

Step 12.7

Add $- 5$ and $2$ .

$- 24 (- \frac{1}{- 3})$

Step 12.8

Cancel the common factor of $3$ .

Tap for more steps...

Step 12.8.1

Move the leading negative in $- \frac{1}{- 3}$ into the numerator.

$- 24 (\frac{- 1}{- 3})$

Step 12.8.2

Factor $3$ out of $- 24$ .

$3 (- 8) \frac{- 1}{- 3}$

Step 12.8.3

Factor $3$ out of $- 3$ .

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

Step 12.8.4

Cancel the common factor.

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

Step 12.8.5

Rewrite the expression.

$- 8 (\frac{- 1}{- 1})$

Step 12.9

Combine $- 8$ and $\frac{- 1}{- 1}$ .

$\frac{- 8 \cdot - 1}{- 1}$

Step 12.10

Multiply $- 8$ by $- 1$ .

$\frac{8}{- 1}$

Step 12.11

Divide $8$ by $- 1$ .

$- 8$