Precalculus Examples

$lim_{x \to 8} \frac{4 - 3 x}{4 x - 9}$

Step 1

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

$\frac{lim_{x \to 8} 4 - 3 x}{lim_{x \to 8} 4 x - 9}$

Step 2

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

$\frac{lim_{x \to 8} 4 - lim_{x \to 8} 3 x}{lim_{x \to 8} 4 x - 9}$

Step 3

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{4 - lim_{x \to 8} 3 x}{lim_{x \to 8} 4 x - 9}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 - 3 lim_{x \to 8} x}{lim_{x \to 8} 4 x - 9}$

Step 5

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

$\frac{4 - 3 lim_{x \to 8} x}{lim_{x \to 8} 4 x - lim_{x \to 8} 9}$

Step 6

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

$\frac{4 - 3 lim_{x \to 8} x}{4 lim_{x \to 8} x - lim_{x \to 8} 9}$

Step 7

Evaluate the limit of $9$ which is constant as $x$ approaches $8$ .

$\frac{4 - 3 lim_{x \to 8} x}{4 lim_{x \to 8} x - 1 \cdot 9}$

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 $x$ .

$\frac{4 - 3 \cdot 8}{4 lim_{x \to 8} x - 1 \cdot 9}$

Step 8.2

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

$\frac{4 - 3 \cdot 8}{4 \cdot 8 - 1 \cdot 9}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $- 3$ by $8$ .

$\frac{4 - 24}{4 \cdot 8 - 1 \cdot 9}$

Step 9.1.2

Subtract $24$ from $4$ .

$\frac{- 20}{4 \cdot 8 - 1 \cdot 9}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $4$ by $8$ .

$\frac{- 20}{32 - 1 \cdot 9}$

Step 9.2.2

Multiply $- 1$ by $9$ .

$\frac{- 20}{32 - 9}$

Step 9.2.3

Subtract $9$ from $32$ .

$\frac{- 20}{23}$

Step 9.3

Move the negative in front of the fraction.

$- \frac{20}{23}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{20}{23}$

Decimal Form:

$- 0.86956521 \dots$