Precalculus Examples

$lim_{x \to 8} \frac{4 x (\log (x)) + 2 x}{10^{100}}$

Step 1

Move the term $\frac{1}{10^{100}}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{10^{100}} lim_{x \to 8} 4 x \log (x) + 2 x$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{1}{10^{100}} (lim_{x \to 8} 4 x \log (x) + lim_{x \to 8} 2 x)$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{10^{100}} (4 lim_{x \to 8} x \log (x) + lim_{x \to 8} 2 x)$

Step 4

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{1}{10^{100}} (4 lim_{x \to 8} x \cdot lim_{x \to 8} \log (x) + lim_{x \to 8} 2 x)$

Step 5

Move the limit inside the logarithm.

$\frac{1}{10^{100}} (4 lim_{x \to 8} x \cdot \log (lim_{x \to 8} x) + lim_{x \to 8} 2 x)$

Step 6

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

$\frac{1}{10^{100}} (4 lim_{x \to 8} x \cdot \log (lim_{x \to 8} x) + 2 lim_{x \to 8} x)$

Step 7

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 7.1

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

$\frac{1}{10^{100}} (4 \cdot 8 \cdot \log (lim_{x \to 8} x) + 2 lim_{x \to 8} x)$

Step 7.2

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

$\frac{1}{10^{100}} (4 \cdot 8 \cdot \log (8) + 2 lim_{x \to 8} x)$

Step 7.3

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

$\frac{1}{10^{100}} (4 \cdot 8 \cdot \log (8) + 2 \cdot 8)$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $4$ by $8$ .

$\frac{1}{10^{100}} (32 \cdot \log (8) + 2 \cdot 8)$

Step 8.1.2

Multiply $2$ by $8$ .

$\frac{1}{10^{100}} (32 \log (8) + 16)$

Step 8.2

Apply the distributive property.

$\frac{1}{10^{100}} (32 \log (8)) + \frac{1}{10^{100}} \cdot 16$

Step 8.3

Multiply $\frac{1}{10^{100}} (32 \log (8))$ .

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Step 8.3.1

Combine $32$ and $\frac{1}{10^{100}}$ .

$\frac{32}{10^{100}} \log (8) + \frac{1}{10^{100}} \cdot 16$

Step 8.3.2

Combine $\frac{32}{10^{100}}$ and $\log (8)$ .

$\frac{32 \log (8)}{10^{100}} + \frac{1}{10^{100}} \cdot 16$

Step 8.4

Combine $\frac{1}{10^{100}}$ and $16$ .

$\frac{32 \log (8)}{10^{100}} + \frac{16}{10^{100}}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{32 \log (8)}{10^{100}} + \frac{16}{10^{100}}$

Decimal Form:

$4.48988795 \cdot 10^{- 99}$