Calculus Examples

$lim_{x \to 8} \frac{8 x + 3}{1 - 2 x}$

Step 1

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

$\frac{lim_{x \to 8} 8 x + 3}{lim_{x \to 8} 1 - 2 x}$

Step 2

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

$\frac{lim_{x \to 8} 8 x + lim_{x \to 8} 3}{lim_{x \to 8} 1 - 2 x}$

Step 3

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{8 lim_{x \to 8} x + lim_{x \to 8} 3}{lim_{x \to 8} 1 - 2 x}$

Step 4

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

$\frac{8 lim_{x \to 8} x + 3}{lim_{x \to 8} 1 - 2 x}$

Step 5

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

$\frac{8 lim_{x \to 8} x + 3}{lim_{x \to 8} 1 - lim_{x \to 8} 2 x}$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{8 lim_{x \to 8} x + 3}{1 - lim_{x \to 8} 2 x}$

Step 7

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

$\frac{8 lim_{x \to 8} x + 3}{1 - 2 lim_{x \to 8} x}$

Step 8

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

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Step 8.1

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

$\frac{8 \cdot 8 + 3}{1 - 2 lim_{x \to 8} x}$

Step 8.2

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

$\frac{8 \cdot 8 + 3}{1 - 2 \cdot 8}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $8$ by $8$ .

$\frac{64 + 3}{1 - 2 \cdot 8}$

Step 9.1.2

Add $64$ and $3$ .

$\frac{67}{1 - 2 \cdot 8}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $- 2$ by $8$ .

$\frac{67}{1 - 16}$

Step 9.2.2

Subtract $16$ from $1$ .

$\frac{67}{- 15}$

Step 9.3

Move the negative in front of the fraction.

$- \frac{67}{15}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{67}{15}$

Decimal Form:

$- 4.4\overline{6}$