Calculus Examples

$lim_{x \to 3} \frac{5 x^{2} + 14 x - 3}{x^{2} + 3 x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} 5 x^{2} + 14 x - 3}{lim_{x \to 3} x^{2} + 3 x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} 5 x^{2} + lim_{x \to 3} 14 x - lim_{x \to 3} 3}{lim_{x \to 3} x^{2} + 3 x}$

Step 3

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

$\frac{5 lim_{x \to 3} x^{2} + lim_{x \to 3} 14 x - lim_{x \to 3} 3}{lim_{x \to 3} x^{2} + 3 x}$

Step 4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{5 {(lim_{x \to 3} x)}^{2} + lim_{x \to 3} 14 x - lim_{x \to 3} 3}{lim_{x \to 3} x^{2} + 3 x}$

Step 5

Move the term $14$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 {(lim_{x \to 3} x)}^{2} + 14 lim_{x \to 3} x - lim_{x \to 3} 3}{lim_{x \to 3} x^{2} + 3 x}$

Step 6

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

$\frac{5 {(lim_{x \to 3} x)}^{2} + 14 lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} x^{2} + 3 x}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{5 {(lim_{x \to 3} x)}^{2} + 14 lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} x^{2} + lim_{x \to 3} 3 x}$

Step 8

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{5 {(lim_{x \to 3} x)}^{2} + 14 lim_{x \to 3} x - 1 \cdot 3}{{(lim_{x \to 3} x)}^{2} + lim_{x \to 3} 3 x}$

Step 9

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

$\frac{5 {(lim_{x \to 3} x)}^{2} + 14 lim_{x \to 3} x - 1 \cdot 3}{{(lim_{x \to 3} x)}^{2} + 3 lim_{x \to 3} x}$

Step 10

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{5 \cdot 3^{2} + 14 lim_{x \to 3} x - 1 \cdot 3}{{(lim_{x \to 3} x)}^{2} + 3 lim_{x \to 3} x}$

Step 10.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{5 \cdot 3^{2} + 14 \cdot 3 - 1 \cdot 3}{{(lim_{x \to 3} x)}^{2} + 3 lim_{x \to 3} x}$

Step 10.3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{5 \cdot 3^{2} + 14 \cdot 3 - 1 \cdot 3}{3^{2} + 3 lim_{x \to 3} x}$

Step 10.4

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{5 \cdot 3^{2} + 14 \cdot 3 - 1 \cdot 3}{3^{2} + 3 \cdot 3}$

Step 11

Simplify the answer.

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Step 11.1

Cancel the common factor of $5 \cdot 3^{2} + 14 \cdot 3 - 1 \cdot 3$ and $3^{2} + 3 \cdot 3$ .

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Step 11.1.1

Factor $- 1$ out of $5 \cdot 3^{2}$ .

$\frac{- (- 5 \cdot 3^{2}) + 14 \cdot 3 - 1 \cdot 3}{3^{2} + 3 \cdot 3}$

Step 11.1.2

Factor $- 1$ out of $14 \cdot 3$ .

$\frac{- (- 5 \cdot 3^{2}) - (- 14 \cdot 3) - 1 \cdot 3}{3^{2} + 3 \cdot 3}$

Step 11.1.3

Factor $- 1$ out of $- (- 5 \cdot 3^{2}) - (- 14 \cdot 3)$ .

$\frac{- (- 5 \cdot 3^{2} - 14 \cdot 3) - 1 \cdot 3}{3^{2} + 3 \cdot 3}$

Step 11.1.4

Factor $- 1$ out of $- 1 \cdot 3$ .

$\frac{- (- 5 \cdot 3^{2} - 14 \cdot 3) - 1 (3)}{3^{2} + 3 \cdot 3}$

Step 11.1.5

Factor $- 1$ out of $- (- 5 \cdot 3^{2} - 14 \cdot 3) - 1 (3)$ .

$\frac{- (- 5 \cdot 3^{2} - 14 \cdot 3 + 3)}{3^{2} + 3 \cdot 3}$

Step 11.1.6

Rewrite $- (- 5 \cdot 3^{2} - 14 \cdot 3 + 3)$ as $- 1 (- 5 \cdot 3^{2} - 14 \cdot 3 + 3)$ .

$\frac{- 1 (- 5 \cdot 3^{2} - 14 \cdot 3 + 3)}{3^{2} + 3 \cdot 3}$

Step 11.1.7

Factor $3$ out of $- 1 (- 5 \cdot 3^{2} - 14 \cdot 3 + 3)$ .

$\frac{3 (- 1 (- 5 \cdot 3 - 14 + 1))}{3^{2} + 3 \cdot 3}$

Step 11.1.8

Cancel the common factors.

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Step 11.1.8.1

Factor $3$ out of $3^{2}$ .

$\frac{3 (- 1 (- 5 \cdot 3 - 14 + 1))}{3 \cdot 3 + 3 \cdot 3}$

Step 11.1.8.2

Factor $3$ out of $3 \cdot 3$ .

$\frac{3 (- 1 (- 5 \cdot 3 - 14 + 1))}{3 \cdot 3 + 3 (3)}$

Step 11.1.8.3

Factor $3$ out of $3 \cdot 3 + 3 (3)$ .

$\frac{3 (- 1 (- 5 \cdot 3 - 14 + 1))}{3 \cdot (3 + 3)}$

Step 11.1.8.4

Cancel the common factor.

$\frac{3 (- 1 (- 5 \cdot 3 - 14 + 1))}{3 \cdot (3 + 3)}$

Step 11.1.8.5

Rewrite the expression.

$\frac{- 1 (- 5 \cdot 3 - 14 + 1)}{3 + 3}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

Multiply $- 5$ by $3$ .

$\frac{- 1 (- 15 - 14 + 1)}{3 + 3}$

Step 11.2.2

Subtract $14$ from $- 15$ .

$\frac{- 1 (- 29 + 1)}{3 + 3}$

Step 11.2.3

Add $- 29$ and $1$ .

$\frac{- 1 \cdot - 28}{3 + 3}$

Step 11.3

Add $3$ and $3$ .

$\frac{- 1 \cdot - 28}{6}$

Step 11.4

Multiply $- 1$ by $- 28$ .

$\frac{28}{6}$

Step 11.5

Cancel the common factor of $28$ and $6$ .

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Step 11.5.1

Factor $2$ out of $28$ .

$\frac{2 (14)}{6}$

Step 11.5.2

Cancel the common factors.

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Step 11.5.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 14}{2 \cdot 3}$

Step 11.5.2.2

Cancel the common factor.

$\frac{2 \cdot 14}{2 \cdot 3}$

Step 11.5.2.3

Rewrite the expression.

$\frac{14}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{14}{3}$

Decimal Form:

$4.\overline{6}$