Calculus Examples

$\frac{lim_{x \to 0} 6 x^{5} - 8 x^{3}}{9 x^{3} - 6 x^{5}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 6 x^{5} - lim_{x \to 0} 8 x^{3}}{9 x^{3} - 6 x^{5}}$

Step 2

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

$\frac{6 lim_{x \to 0} x^{5} - lim_{x \to 0} 8 x^{3}}{9 x^{3} - 6 x^{5}}$

Step 3

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{6 {(lim_{x \to 0} x)}^{5} - lim_{x \to 0} 8 x^{3}}{9 x^{3} - 6 x^{5}}$

Step 4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 {(lim_{x \to 0} x)}^{5} - 8 lim_{x \to 0} x^{3}}{9 x^{3} - 6 x^{5}}$

Step 5

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

$\frac{6 {(lim_{x \to 0} x)}^{5} - 8 {(lim_{x \to 0} x)}^{3}}{9 x^{3} - 6 x^{5}}$

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{6 \cdot 0^{5} - 8 {(lim_{x \to 0} x)}^{3}}{9 x^{3} - 6 x^{5}}$

Step 6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{6 \cdot 0^{5} - 8 \cdot 0^{3}}{9 x^{3} - 6 x^{5}}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{6 \cdot 0 - 8 \cdot 0^{3}}{9 x^{3} - 6 x^{5}}$

Step 7.1.2

Multiply $6$ by $0$ .

$\frac{0 - 8 \cdot 0^{3}}{9 x^{3} - 6 x^{5}}$

Step 7.1.3

Raising $0$ to any positive power yields $0$ .

$\frac{0 - 8 \cdot 0}{9 x^{3} - 6 x^{5}}$

Step 7.1.4

Multiply $- 8$ by $0$ .

$\frac{0 + 0}{9 x^{3} - 6 x^{5}}$

Step 7.1.5

Add $0$ and $0$ .

$\frac{0}{9 x^{3} - 6 x^{5}}$

Step 7.2

Factor $3 x^{3}$ out of $9 x^{3} - 6 x^{5}$ .

Tap for more steps...

Step 7.2.1

Factor $3 x^{3}$ out of $9 x^{3}$ .

$\frac{0}{3 x^{3} (3) - 6 x^{5}}$

Step 7.2.2

Factor $3 x^{3}$ out of $- 6 x^{5}$ .

$\frac{0}{3 x^{3} (3) + 3 x^{3} (- 2 x^{2})}$

Step 7.2.3

Factor $3 x^{3}$ out of $3 x^{3} (3) + 3 x^{3} (- 2 x^{2})$ .

$\frac{0}{3 x^{3} (3 - 2 x^{2})}$

Step 7.3

Cancel the common factor of $0$ and $3$ .

Tap for more steps...

Step 7.3.1

Factor $3$ out of $0$ .

$\frac{3 (0)}{3 x^{3} (3 - 2 x^{2})}$

Step 7.3.2

Cancel the common factors.

Tap for more steps...

Step 7.3.2.1

Factor $3$ out of $3 x^{3} (3 - 2 x^{2})$ .

$\frac{3 (0)}{3 (x^{3} (3 - 2 x^{2}))}$

Step 7.3.2.2

Cancel the common factor.

$\frac{3 \cdot 0}{3 (x^{3} (3 - 2 x^{2}))}$

Step 7.3.2.3

Rewrite the expression.

$\frac{0}{x^{3} (3 - 2 x^{2})}$

Step 7.4

Divide $0$ by $x^{3} (3 - 2 x^{2})$ .

$0$