Calculus Examples

$lim_{h \to 9} {(5 (9 + h))}^{3} - {(9 + h)}^{2} + 6$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $9$ .

$lim_{h \to 9} {(5 (9 + h))}^{3} - lim_{h \to 9} {(9 + h)}^{2} + lim_{h \to 9} 6$

Step 2

Move the exponent $3$ from ${(5 (9 + h))}^{3}$ outside the limit using the Limits Power Rule.

${(lim_{h \to 9} 5 (9 + h))}^{3} - lim_{h \to 9} {(9 + h)}^{2} + lim_{h \to 9} 6$

Step 3

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

${(5 lim_{h \to 9} 9 + h)}^{3} - lim_{h \to 9} {(9 + h)}^{2} + lim_{h \to 9} 6$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $9$ .

${(5 (lim_{h \to 9} 9 + lim_{h \to 9} h))}^{3} - lim_{h \to 9} {(9 + h)}^{2} + lim_{h \to 9} 6$

Step 5

Evaluate the limit of $9$ which is constant as $h$ approaches $9$ .

${(5 (9 + lim_{h \to 9} h))}^{3} - lim_{h \to 9} {(9 + h)}^{2} + lim_{h \to 9} 6$

Step 6

Move the exponent $2$ from ${(9 + h)}^{2}$ outside the limit using the Limits Power Rule.

${(5 (9 + lim_{h \to 9} h))}^{3} - {(lim_{h \to 9} 9 + h)}^{2} + lim_{h \to 9} 6$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $9$ .

${(5 (9 + lim_{h \to 9} h))}^{3} - {(lim_{h \to 9} 9 + lim_{h \to 9} h)}^{2} + lim_{h \to 9} 6$

Step 8

Evaluate the limit of $9$ which is constant as $h$ approaches $9$ .

${(5 (9 + lim_{h \to 9} h))}^{3} - {(9 + lim_{h \to 9} h)}^{2} + lim_{h \to 9} 6$

Step 9

Evaluate the limit of $6$ which is constant as $h$ approaches $9$ .

${(5 (9 + lim_{h \to 9} h))}^{3} - {(9 + lim_{h \to 9} h)}^{2} + 6$

Step 10

Evaluate the limits by plugging in $9$ for all occurrences of $h$ .

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Step 10.1

Evaluate the limit of $h$ by plugging in $9$ for $h$ .

${(5 (9 + 9))}^{3} - {(9 + lim_{h \to 9} h)}^{2} + 6$

Step 10.2

Evaluate the limit of $h$ by plugging in $9$ for $h$ .

${(5 (9 + 9))}^{3} - {(9 + 9)}^{2} + 6$

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

Add $9$ and $9$ .

${(5 \cdot 18)}^{3} - {(9 + 9)}^{2} + 6$

Step 11.1.2

Multiply $5$ by $18$ .

$90^{3} - {(9 + 9)}^{2} + 6$

Step 11.1.3

Raise $90$ to the power of $3$ .

$729000 - {(9 + 9)}^{2} + 6$

Step 11.1.4

Add $9$ and $9$ .

$729000 - 18^{2} + 6$

Step 11.1.5

Raise $18$ to the power of $2$ .

$729000 - 1 \cdot 324 + 6$

Step 11.1.6

Multiply $- 1$ by $324$ .

$729000 - 324 + 6$

Step 11.2

Subtract $324$ from $729000$ .

$728676 + 6$

Step 11.3

Add $728676$ and $6$ .

$728682$