Calculus Examples

$\frac{lim_{h \to o} {(7 + h)}^{3} - 343}{h}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{h \to o} {(7 + h)}^{3} - lim_{h \to o} 343}{h}$

Step 1.2

Move the exponent $3$ from ${(7 + h)}^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{h \to o} 7 + h)}^{3} - lim_{h \to o} 343}{h}$

Step 1.3

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

$\frac{{(lim_{h \to o} 7 + lim_{h \to o} h)}^{3} - lim_{h \to o} 343}{h}$

Step 1.4

Evaluate the limit of $7$ which is constant as $h$ approaches $o$ .

$\frac{{(7 + lim_{h \to o} h)}^{3} - lim_{h \to o} 343}{h}$

Step 1.5

Evaluate the limit of $343$ which is constant as $h$ approaches $o$ .

$\frac{{(7 + lim_{h \to o} h)}^{3} - 1 \cdot 343}{h}$

Step 2

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

$\frac{{(7 + o)}^{3} - 1 \cdot 343}{h}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $343$ as $7^{3}$ .

$\frac{{(7 + o)}^{3} - 7^{3}}{h}$

Step 3.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 7 + o$ and $b = 7$ .

$\frac{(7 + o - 7) ({(7 + o)}^{2} + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3

Simplify.

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Step 3.3.1

Subtract $7$ from $7$ .

$\frac{(o + 0) ({(7 + o)}^{2} + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3.2

Add $o$ and $0$ .

$\frac{o ({(7 + o)}^{2} + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3.3

Rewrite ${(7 + o)}^{2}$ as $(7 + o) (7 + o)$ .

$\frac{o ((7 + o) (7 + o) + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3.4

Expand $(7 + o) (7 + o)$ using the FOIL Method.

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Step 3.3.4.1

Apply the distributive property.

$\frac{o (7 (7 + o) + o (7 + o) + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3.4.2

Apply the distributive property.

$\frac{o (7 \cdot 7 + 7 o + o (7 + o) + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3.4.3

Apply the distributive property.

$\frac{o (7 \cdot 7 + 7 o + o \cdot 7 + o \cdot o + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3.5

Simplify and combine like terms.

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Step 3.3.5.1

Simplify each term.

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Step 3.3.5.1.1

Multiply $7$ by $7$ .

$\frac{o (49 + 7 o + o \cdot 7 + o \cdot o + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3.5.1.2

Move $7$ to the left of $o$ .

$\frac{o (49 + 7 o + 7 \cdot o + o \cdot o + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3.5.1.3

Multiply $o$ by $o$ .

$\frac{o (49 + 7 o + 7 o + o^{2} + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3.5.2

Add $7 o$ and $7 o$ .

$\frac{o (49 + 14 o + o^{2} + (7 + o) \cdot 7 + 7^{2})}{h}$

Step 3.3.6

Apply the distributive property.

$\frac{o (49 + 14 o + o^{2} + 7 \cdot 7 + o \cdot 7 + 7^{2})}{h}$

Step 3.3.7

Multiply $7$ by $7$ .

$\frac{o (49 + 14 o + o^{2} + 49 + o \cdot 7 + 7^{2})}{h}$

Step 3.3.8

Move $7$ to the left of $o$ .

$\frac{o (49 + 14 o + o^{2} + 49 + 7 \cdot o + 7^{2})}{h}$

Step 3.3.9

Raise $7$ to the power of $2$ .

$\frac{o (49 + 14 o + o^{2} + 49 + 7 o + 49)}{h}$

Step 3.3.10

Add $49$ and $49$ .

$\frac{o (14 o + o^{2} + 98 + 7 o + 49)}{h}$

Step 3.3.11

Add $14 o$ and $7 o$ .

$\frac{o (21 o + o^{2} + 98 + 49)}{h}$

Step 3.3.12

Add $98$ and $49$ .

$\frac{o (21 o + o^{2} + 147)}{h}$

Step 3.3.13

Reorder terms.

$\frac{o (o^{2} + 21 o + 147)}{h}$