Calculus Examples

$lim_{x \to 8} \frac{4 \cdot 3^{20} + 2^{19} - 4}{5 \cdot 3^{19} + 2^{21} + 1}$

Step 1

Evaluate the limit of $\frac{4 \cdot 3^{20} + 2^{19} - 4}{5 \cdot 3^{19} + 2^{21} + 1}$ which is constant as $x$ approaches $8$ .

$\frac{4 \cdot 3^{20} + 2^{19} - 4}{5 \cdot 3^{19} + 2^{21} + 1}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Raise $3$ to the power of $20$ .

$\frac{4 \cdot 3486784401 + 2^{19} - 4}{5 \cdot 3^{19} + 2^{21} + 1}$

Step 2.1.2

Multiply $4$ by $3486784401$ .

$\frac{13947137604 + 2^{19} - 4}{5 \cdot 3^{19} + 2^{21} + 1}$

Step 2.1.3

Raise $2$ to the power of $19$ .

$\frac{13947137604 + 524288 - 4}{5 \cdot 3^{19} + 2^{21} + 1}$

Step 2.1.4

Add $13947137604$ and $524288$ .

$\frac{13947661892 - 4}{5 \cdot 3^{19} + 2^{21} + 1}$

Step 2.1.5

Subtract $4$ from $13947661892$ .

$\frac{13947661888}{5 \cdot 3^{19} + 2^{21} + 1}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Raise $3$ to the power of $19$ .

$\frac{13947661888}{5 \cdot 1162261467 + 2^{21} + 1}$

Step 2.2.2

Multiply $5$ by $1162261467$ .

$\frac{13947661888}{5811307335 + 2^{21} + 1}$

Step 2.2.3

Raise $2$ to the power of $21$ .

$\frac{13947661888}{5811307335 + 2097152 + 1}$

Step 2.2.4

Add $5811307335$ and $2097152$ .

$\frac{13947661888}{5813404487 + 1}$

Step 2.2.5

Add $5813404487$ and $1$ .

$\frac{13947661888}{5813404488}$

Step 2.3

Cancel the common factor of $13947661888$ and $5813404488$ .

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Step 2.3.1

Factor $8$ out of $13947661888$ .

$\frac{8 (1743457736)}{5813404488}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $8$ out of $5813404488$ .

$\frac{8 \cdot 1743457736}{8 \cdot 726675561}$

Step 2.3.2.2

Cancel the common factor.

$\frac{8 \cdot 1743457736}{8 \cdot 726675561}$

Step 2.3.2.3

Rewrite the expression.

$\frac{1743457736}{726675561}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{1743457736}{726675561}$

Decimal Form:

$2.39922439 \dots$