Calculus Examples

$lim_{x \to 8} 3^{- 2 x} + 3$

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 $x$ approaches $8$ .

$lim_{x \to 8} 3^{- 2 x} + lim_{x \to 8} 3$

Step 1.2

Move the limit into the exponent.

$3^{lim_{x \to 8} - 2 x} + lim_{x \to 8} 3$

Step 1.3

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

$3^{- 2 lim_{x \to 8} x} + lim_{x \to 8} 3$

Step 1.4

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$3^{- 2 lim_{x \to 8} x} + 3$

Step 2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$3^{- 2 \cdot 8} + 3$

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $- 2$ by $8$ .

$3^{- 16} + 3$

Step 3.1.2

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

$\frac{1}{3^{16}} + 3$

Step 3.1.3

Raise $3$ to the power of $16$ .

$\frac{1}{43046721} + 3$

Step 3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{43046721}{43046721}$ .

$\frac{1}{43046721} + 3 \cdot \frac{43046721}{43046721}$

Step 3.3

Combine $3$ and $\frac{43046721}{43046721}$ .

$\frac{1}{43046721} + \frac{3 \cdot 43046721}{43046721}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{1 + 3 \cdot 43046721}{43046721}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $3$ by $43046721$ .

$\frac{1 + 129140163}{43046721}$

Step 3.5.2

Add $1$ and $129140163$ .

$\frac{129140164}{43046721}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{129140164}{43046721}$

Decimal Form:

$3.00000002 \dots$