Calculus Examples

$lim_{x \to \frac{5}{3}} \frac{x - 7}{3 x + 5}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{5}{3}$ .

$\frac{lim_{x \to \frac{5}{3}} x - 7}{lim_{x \to \frac{5}{3}} 3 x + 5}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{5}{3}$ .

$\frac{lim_{x \to \frac{5}{3}} x - lim_{x \to \frac{5}{3}} 7}{lim_{x \to \frac{5}{3}} 3 x + 5}$

Step 3

Evaluate the limit of $7$ which is constant as $x$ approaches $\frac{5}{3}$ .

$\frac{lim_{x \to \frac{5}{3}} x - 1 \cdot 7}{lim_{x \to \frac{5}{3}} 3 x + 5}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{5}{3}$ .

$\frac{lim_{x \to \frac{5}{3}} x - 1 \cdot 7}{lim_{x \to \frac{5}{3}} 3 x + lim_{x \to \frac{5}{3}} 5}$

Step 5

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

$\frac{lim_{x \to \frac{5}{3}} x - 1 \cdot 7}{3 lim_{x \to \frac{5}{3}} x + lim_{x \to \frac{5}{3}} 5}$

Step 6

Evaluate the limit of $5$ which is constant as $x$ approaches $\frac{5}{3}$ .

$\frac{lim_{x \to \frac{5}{3}} x - 1 \cdot 7}{3 lim_{x \to \frac{5}{3}} x + 5}$

Step 7

Evaluate the limits by plugging in $\frac{5}{3}$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $\frac{5}{3}$ for $x$ .

$\frac{\frac{5}{3} - 1 \cdot 7}{3 lim_{x \to \frac{5}{3}} x + 5}$

Step 7.2

Evaluate the limit of $x$ by plugging in $\frac{5}{3}$ for $x$ .

$\frac{\frac{5}{3} - 1 \cdot 7}{3 (\frac{5}{3}) + 5}$

Step 8

Simplify the answer.

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Step 8.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 8.1.1

Multiply $\frac{\frac{5}{3} - 1 \cdot 7}{3 (\frac{5}{3}) + 5}$ by $\frac{3}{3}$ .

$\frac{3}{3} \cdot \frac{\frac{5}{3} - 1 \cdot 7}{3 (\frac{5}{3}) + 5}$

Step 8.1.2

Combine.

$\frac{3 (\frac{5}{3} - 1 \cdot 7)}{3 (3 (\frac{5}{3}) + 5)}$

Step 8.2

Apply the distributive property.

$\frac{3 (\frac{5}{3}) + 3 (- 1 \cdot 7)}{3 (3 (\frac{5}{3})) + 3 \cdot 5}$

Step 8.3

Cancel the common factor of $3$ .

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Step 8.3.1

Cancel the common factor.

$\frac{3 (\frac{5}{3}) + 3 (- 1 \cdot 7)}{3 (3 (\frac{5}{3})) + 3 \cdot 5}$

Step 8.3.2

Rewrite the expression.

$\frac{5 + 3 (- 1 \cdot 7)}{3 (3 (\frac{5}{3})) + 3 \cdot 5}$

Step 8.4

Simplify the numerator.

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Step 8.4.1

Multiply $3 (- 1 \cdot 7)$ .

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Step 8.4.1.1

Multiply $- 1$ by $7$ .

$\frac{5 + 3 \cdot - 7}{3 \cdot 3 \frac{5}{3} + 3 \cdot 5}$

Step 8.4.1.2

Multiply $3$ by $- 7$ .

$\frac{5 - 21}{3 \cdot 3 \frac{5}{3} + 3 \cdot 5}$

Step 8.4.2

Subtract $21$ from $5$ .

$\frac{- 16}{3 \cdot 3 \frac{5}{3} + 3 \cdot 5}$

Step 8.5

Simplify the denominator.

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Step 8.5.1

Multiply $3$ by $3$ .

$\frac{- 16}{9 (\frac{5}{3}) + 3 \cdot 5}$

Step 8.5.2

Cancel the common factor of $3$ .

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Step 8.5.2.1

Factor $3$ out of $9$ .

$\frac{- 16}{3 (3) \frac{5}{3} + 3 \cdot 5}$

Step 8.5.2.2

Cancel the common factor.

$\frac{- 16}{3 \cdot 3 \frac{5}{3} + 3 \cdot 5}$

Step 8.5.2.3

Rewrite the expression.

$\frac{- 16}{3 \cdot 5 + 3 \cdot 5}$

Step 8.5.3

Multiply $3$ by $5$ .

$\frac{- 16}{15 + 3 \cdot 5}$

Step 8.5.4

Multiply $3$ by $5$ .

$\frac{- 16}{15 + 15}$

Step 8.5.5

Add $15$ and $15$ .

$\frac{- 16}{30}$

Step 8.6

Cancel the common factor of $- 16$ and $30$ .

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Step 8.6.1

Factor $2$ out of $- 16$ .

$\frac{2 (- 8)}{30}$

Step 8.6.2

Cancel the common factors.

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Step 8.6.2.1

Factor $2$ out of $30$ .

$\frac{2 \cdot - 8}{2 \cdot 15}$

Step 8.6.2.2

Cancel the common factor.

$\frac{2 \cdot - 8}{2 \cdot 15}$

Step 8.6.2.3

Rewrite the expression.

$\frac{- 8}{15}$

Step 8.7

Move the negative in front of the fraction.

$- \frac{8}{15}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{8}{15}$

Decimal Form:

$- 0.5\overline{3}$