Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

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 $x$ .

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

Step 8.2

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

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

Step 9

Simplify the answer.

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Step 9.1

Multiply $3$ by $8$ .

$\frac{p + 24}{p \cdot 8 - 1 \cdot 3}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Move $8$ to the left of $p$ .

$\frac{p + 24}{8 \cdot p - 1 \cdot 3}$

Step 9.2.2

Multiply $- 1$ by $3$ .

$\frac{p + 24}{8 p - 3}$