Calculus Examples

$lim_{x \to 5} (\frac{8^{3 x}}{x - 5}) + \frac{8^{15}}{x - 5}$

Step 1

Combine fractions using a common denominator.

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Step 1.1

Combine the numerators over the common denominator.

$lim_{x \to 5} \frac{8^{3 x} + 8^{15}}{x - 5}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Rewrite $8^{3 x}$ as ${(8^{x})}^{3}$ .

$lim_{x \to 5} \frac{{(8^{x})}^{3} + 8^{15}}{x - 5}$

Step 1.2.2

Rewrite $8^{15}$ as ${(8^{5})}^{3}$ .

$lim_{x \to 5} \frac{{(8^{x})}^{3} + {(8^{5})}^{3}}{x - 5}$

Step 1.2.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 8^{x}$ and $b = 8^{5}$ .

$lim_{x \to 5} \frac{(8^{x} + 8^{5}) ({(8^{x})}^{2} - 8^{x} \cdot 8^{5} + {(8^{5})}^{2})}{x - 5}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Raise $8$ to the power of $5$ .

$lim_{x \to 5} \frac{(8^{x} + 32768) ({(8^{x})}^{2} - 8^{x} \cdot 8^{5} + {(8^{5})}^{2})}{x - 5}$

Step 1.2.4.2

Multiply the exponents in ${(8^{x})}^{2}$ .

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Step 1.2.4.2.1

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

$lim_{x \to 5} \frac{(8^{x} + 32768) (8^{x \cdot 2} - 8^{x} \cdot 8^{5} + {(8^{5})}^{2})}{x - 5}$

Step 1.2.4.2.2

Move $2$ to the left of $x$ .

$lim_{x \to 5} \frac{(8^{x} + 32768) (8^{2 x} - 8^{x} \cdot 8^{5} + {(8^{5})}^{2})}{x - 5}$

Step 1.2.4.3

Multiply $8^{x}$ by $8^{5}$ by adding the exponents.

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Step 1.2.4.3.1

Move $8^{5}$ .

$lim_{x \to 5} \frac{(8^{x} + 32768) (8^{2 x} - (8^{5} \cdot 8^{x}) + {(8^{5})}^{2})}{x - 5}$

Step 1.2.4.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 5} \frac{(8^{x} + 32768) (8^{2 x} - 8^{5 + x} + {(8^{5})}^{2})}{x - 5}$

Step 1.2.4.4

Multiply the exponents in ${(8^{5})}^{2}$ .

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Step 1.2.4.4.1

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

$lim_{x \to 5} \frac{(8^{x} + 32768) (8^{2 x} - 8^{5 + x} + 8^{5 \cdot 2})}{x - 5}$

Step 1.2.4.4.2

Multiply $5$ by $2$ .

$lim_{x \to 5} \frac{(8^{x} + 32768) (8^{2 x} - 8^{5 + x} + 8^{10})}{x - 5}$

Step 1.2.4.5

Raise $8$ to the power of $10$ .

$lim_{x \to 5} \frac{(8^{x} + 32768) (8^{2 x} - 8^{5 + x} + 1073741824)}{x - 5}$

Step 1.2.4.6

Reorder terms.

$lim_{x \to 5} \frac{(8^{x} + 32768) (8^{2 x} + 1073741824 - 8^{5 + x})}{x - 5}$

Step 2

Since the function approaches $- \infty$ from the left and $\infty$ from the right, the limit does not exist.

$Does not exist$