Calculus Examples

$lim_{x \to - \frac{1}{2}} 1 - 5 x^{3}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \frac{1}{2}$ .

$lim_{x \to - \frac{1}{2}} 1 - lim_{x \to - \frac{1}{2}} 5 x^{3}$

Step 1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $- \frac{1}{2}$ .

$1 - lim_{x \to - \frac{1}{2}} 5 x^{3}$

Step 1.3

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

$1 - 5 lim_{x \to - \frac{1}{2}} x^{3}$

Step 1.4

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

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

Step 2

Evaluate the limit of $x$ by plugging in $- \frac{1}{2}$ for $x$ .

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

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 3.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 3.1.1.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 3.1.2

Raise $- 1$ to the power of $3$ .

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

Step 3.1.3

One to any power is one.

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

Step 3.1.4

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

$1 - 5 (- \frac{1}{8})$

Step 3.1.5

Multiply $- 5 (- \frac{1}{8})$ .

Tap for more steps...

Step 3.1.5.1

Multiply $- 1$ by $- 5$ .

$1 + 5 (\frac{1}{8})$

Step 3.1.5.2

Combine $5$ and $\frac{1}{8}$ .

$1 + \frac{5}{8}$

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Add $8$ and $5$ .

$\frac{13}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{8}$

Decimal Form:

$1.625$