Calculus Examples

$lim_{w \to 5} {(2 w^{4} - 9 w^{3} + 19)}^{- \frac{1}{2}}$

Step 1

Move the exponent $- \frac{1}{2}$ from ${(2 w^{4} - 9 w^{3} + 19)}^{- \frac{1}{2}}$ outside the limit using the Limits Power Rule.

${(lim_{w \to 5} 2 w^{4} - 9 w^{3} + 19)}^{- \frac{1}{2}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $w$ approaches $5$ .

${(lim_{w \to 5} 2 w^{4} - lim_{w \to 5} 9 w^{3} + lim_{w \to 5} 19)}^{- \frac{1}{2}}$

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $w$ .

${(2 lim_{w \to 5} w^{4} - lim_{w \to 5} 9 w^{3} + lim_{w \to 5} 19)}^{- \frac{1}{2}}$

Step 4

Move the exponent $4$ from $w^{4}$ outside the limit using the Limits Power Rule.

${(2 {(lim_{w \to 5} w)}^{4} - lim_{w \to 5} 9 w^{3} + lim_{w \to 5} 19)}^{- \frac{1}{2}}$

Step 5

Move the term $9$ outside of the limit because it is constant with respect to $w$ .

${(2 {(lim_{w \to 5} w)}^{4} - 9 lim_{w \to 5} w^{3} + lim_{w \to 5} 19)}^{- \frac{1}{2}}$

Step 6

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

${(2 {(lim_{w \to 5} w)}^{4} - 9 {(lim_{w \to 5} w)}^{3} + lim_{w \to 5} 19)}^{- \frac{1}{2}}$

Step 7

Evaluate the limit of $19$ which is constant as $w$ approaches $5$ .

${(2 {(lim_{w \to 5} w)}^{4} - 9 {(lim_{w \to 5} w)}^{3} + 19)}^{- \frac{1}{2}}$

Step 8

Evaluate the limits by plugging in $5$ for all occurrences of $w$ .

Tap for more steps...

Step 8.1

Evaluate the limit of $w$ by plugging in $5$ for $w$ .

${(2 \cdot 5^{4} - 9 {(lim_{w \to 5} w)}^{3} + 19)}^{- \frac{1}{2}}$

Step 8.2

Evaluate the limit of $w$ by plugging in $5$ for $w$ .

${(2 \cdot 5^{4} - 9 \cdot 5^{3} + 19)}^{- \frac{1}{2}}$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Raise $5$ to the power of $4$ .

${(2 \cdot 625 - 9 \cdot 5^{3} + 19)}^{- \frac{1}{2}}$

Step 9.1.2

Multiply $2$ by $625$ .

${(1250 - 9 \cdot 5^{3} + 19)}^{- \frac{1}{2}}$

Step 9.1.3

Raise $5$ to the power of $3$ .

${(1250 - 9 \cdot 125 + 19)}^{- \frac{1}{2}}$

Step 9.1.4

Multiply $- 9$ by $125$ .

${(1250 - 1125 + 19)}^{- \frac{1}{2}}$

Step 9.2

Subtract $1125$ from $1250$ .

${(125 + 19)}^{- \frac{1}{2}}$

Step 9.3

Add $125$ and $19$ .

$144^{- \frac{1}{2}}$

Step 9.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{144^{\frac{1}{2}}}$

Step 9.5

Simplify the denominator.

Tap for more steps...

Step 9.5.1

Rewrite $144$ as $12^{2}$ .

$\frac{1}{{(12^{2})}^{\frac{1}{2}}}$

Step 9.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{12^{2 (\frac{1}{2})}}$

Step 9.5.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.5.3.1

Cancel the common factor.

$\frac{1}{12^{2 (\frac{1}{2})}}$

Step 9.5.3.2

Rewrite the expression.

$\frac{1}{12^{1}}$

Step 9.5.4

Evaluate the exponent.

$\frac{1}{12}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{12}$

Decimal Form:

$0.08\overline{3}$