Precalculus Examples

$lim_{x \to 8} \frac{2 x + 3}{5 x + 1}$

Step 1

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

$\frac{lim_{x \to 8} 2 x + 3}{lim_{x \to 8} 5 x + 1}$

Step 2

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

$\frac{lim_{x \to 8} 2 x + lim_{x \to 8} 3}{lim_{x \to 8} 5 x + 1}$

Step 3

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

$\frac{2 lim_{x \to 8} x + lim_{x \to 8} 3}{lim_{x \to 8} 5 x + 1}$

Step 4

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

$\frac{2 lim_{x \to 8} x + 3}{lim_{x \to 8} 5 x + 1}$

Step 5

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

$\frac{2 lim_{x \to 8} x + 3}{lim_{x \to 8} 5 x + lim_{x \to 8} 1}$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 8} x + 3}{5 lim_{x \to 8} x + lim_{x \to 8} 1}$

Step 7

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

$\frac{2 lim_{x \to 8} x + 3}{5 lim_{x \to 8} x + 1}$

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 $x$ .

$\frac{2 \cdot 8 + 3}{5 lim_{x \to 8} x + 1}$

Step 8.2

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

$\frac{2 \cdot 8 + 3}{5 \cdot 8 + 1}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $2$ by $8$ .

$\frac{16 + 3}{5 \cdot 8 + 1}$

Step 9.1.2

Add $16$ and $3$ .

$\frac{19}{5 \cdot 8 + 1}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $5$ by $8$ .

$\frac{19}{40 + 1}$

Step 9.2.2

Add $40$ and $1$ .

$\frac{19}{41}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{19}{41}$

Decimal Form:

$0.\overline{46341}$