Calculus Examples

$lim_{x \to \frac{1}{3}} \frac{3 x^{2} + 5 x - 2}{6 x^{2} + x - 1}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to \frac{1}{3}} 3 x^{2} + 5 x - 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{3}$ .

$\frac{lim_{x \to \frac{1}{3}} 3 x^{2} + lim_{x \to \frac{1}{3}} 5 x - lim_{x \to \frac{1}{3}} 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.2

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

$\frac{3 lim_{x \to \frac{1}{3}} x^{2} + lim_{x \to \frac{1}{3}} 5 x - lim_{x \to \frac{1}{3}} 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.3

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

$\frac{3 {(lim_{x \to \frac{1}{3}} x)}^{2} + lim_{x \to \frac{1}{3}} 5 x - lim_{x \to \frac{1}{3}} 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.4

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

$\frac{3 {(lim_{x \to \frac{1}{3}} x)}^{2} + 5 lim_{x \to \frac{1}{3}} x - lim_{x \to \frac{1}{3}} 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.5

Evaluate the limit of $2$ which is constant as $x$ approaches $\frac{1}{3}$ .

$\frac{3 {(lim_{x \to \frac{1}{3}} x)}^{2} + 5 lim_{x \to \frac{1}{3}} x - 1 \cdot 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.6

Evaluate the limits by plugging in $\frac{1}{3}$ for all occurrences of $x$ .

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Step 1.1.2.6.1

Evaluate the limit of $x$ by plugging in $\frac{1}{3}$ for $x$ .

$\frac{3 {(\frac{1}{3})}^{2} + 5 lim_{x \to \frac{1}{3}} x - 1 \cdot 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.6.2

Evaluate the limit of $x$ by plugging in $\frac{1}{3}$ for $x$ .

$\frac{3 {(\frac{1}{3})}^{2} + 5 (\frac{1}{3}) - 1 \cdot 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Simplify each term.

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Step 1.1.2.7.1.1

Apply the product rule to $\frac{1}{3}$ .

$\frac{3 \frac{1^{2}}{3^{2}} + 5 (\frac{1}{3}) - 1 \cdot 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7.1.2

Cancel the common factor of $3$ .

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Step 1.1.2.7.1.2.1

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

$\frac{3 \frac{1^{2}}{3 \cdot 3} + 5 (\frac{1}{3}) - 1 \cdot 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7.1.2.2

Cancel the common factor.

$\frac{3 \frac{1^{2}}{3 \cdot 3} + 5 (\frac{1}{3}) - 1 \cdot 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7.1.2.3

Rewrite the expression.

$\frac{\frac{1^{2}}{3} + 5 (\frac{1}{3}) - 1 \cdot 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7.1.3

One to any power is one.

$\frac{\frac{1}{3} + 5 (\frac{1}{3}) - 1 \cdot 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7.1.4

Combine $5$ and $\frac{1}{3}$ .

$\frac{\frac{1}{3} + \frac{5}{3} - 1 \cdot 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7.1.5

Multiply $- 1$ by $2$ .

$\frac{\frac{1}{3} + \frac{5}{3} - 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7.2

Combine the numerators over the common denominator.

$\frac{- 2 + \frac{1 + 5}{3}}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7.3

Add $1$ and $5$ .

$\frac{- 2 + \frac{6}{3}}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7.4

Divide $6$ by $3$ .

$\frac{- 2 + 2}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.2.7.5

Add $- 2$ and $2$ .

$\frac{0}{lim_{x \to \frac{1}{3}} 6 x^{2} + x - 1}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{3}$ .

$\frac{0}{lim_{x \to \frac{1}{3}} 6 x^{2} + lim_{x \to \frac{1}{3}} x - lim_{x \to \frac{1}{3}} 1}$

Step 1.1.3.2

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{6 lim_{x \to \frac{1}{3}} x^{2} + lim_{x \to \frac{1}{3}} x - lim_{x \to \frac{1}{3}} 1}$

Step 1.1.3.3

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

$\frac{0}{6 {(lim_{x \to \frac{1}{3}} x)}^{2} + lim_{x \to \frac{1}{3}} x - lim_{x \to \frac{1}{3}} 1}$

Step 1.1.3.4

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{1}{3}$ .

$\frac{0}{6 {(lim_{x \to \frac{1}{3}} x)}^{2} + lim_{x \to \frac{1}{3}} x - 1 \cdot 1}$

Step 1.1.3.5

Evaluate the limits by plugging in $\frac{1}{3}$ for all occurrences of $x$ .

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Step 1.1.3.5.1

Evaluate the limit of $x$ by plugging in $\frac{1}{3}$ for $x$ .

$\frac{0}{6 {(\frac{1}{3})}^{2} + lim_{x \to \frac{1}{3}} x - 1 \cdot 1}$

Step 1.1.3.5.2

Evaluate the limit of $x$ by plugging in $\frac{1}{3}$ for $x$ .

$\frac{0}{6 {(\frac{1}{3})}^{2} + \frac{1}{3} - 1 \cdot 1}$

Step 1.1.3.6

Simplify the answer.

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Step 1.1.3.6.1

Simplify each term.

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Step 1.1.3.6.1.1

Apply the product rule to $\frac{1}{3}$ .

$\frac{0}{6 \frac{1^{2}}{3^{2}} + \frac{1}{3} - 1 \cdot 1}$

Step 1.1.3.6.1.2

One to any power is one.

$\frac{0}{6 \frac{1}{3^{2}} + \frac{1}{3} - 1 \cdot 1}$

Step 1.1.3.6.1.3

Raise $3$ to the power of $2$ .

$\frac{0}{6 (\frac{1}{9}) + \frac{1}{3} - 1 \cdot 1}$

Step 1.1.3.6.1.4

Cancel the common factor of $3$ .

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Step 1.1.3.6.1.4.1

Factor $3$ out of $6$ .

$\frac{0}{3 (2) \frac{1}{9} + \frac{1}{3} - 1 \cdot 1}$

Step 1.1.3.6.1.4.2

Factor $3$ out of $9$ .

$\frac{0}{3 \cdot 2 \frac{1}{3 \cdot 3} + \frac{1}{3} - 1 \cdot 1}$

Step 1.1.3.6.1.4.3

Cancel the common factor.

$\frac{0}{3 \cdot 2 \frac{1}{3 \cdot 3} + \frac{1}{3} - 1 \cdot 1}$

Step 1.1.3.6.1.4.4

Rewrite the expression.

$\frac{0}{2 (\frac{1}{3}) + \frac{1}{3} - 1 \cdot 1}$

Step 1.1.3.6.1.5

Combine $2$ and $\frac{1}{3}$ .

$\frac{0}{\frac{2}{3} + \frac{1}{3} - 1 \cdot 1}$

Step 1.1.3.6.1.6

Multiply $- 1$ by $1$ .

$\frac{0}{\frac{2}{3} + \frac{1}{3} - 1}$

Step 1.1.3.6.2

Combine the numerators over the common denominator.

$\frac{0}{- 1 + \frac{2 + 1}{3}}$

Step 1.1.3.6.3

Add $2$ and $1$ .

$\frac{0}{- 1 + \frac{3}{3}}$

Step 1.1.3.6.4

Divide $3$ by $3$ .

$\frac{0}{- 1 + 1}$

Step 1.1.3.6.5

Add $- 1$ and $1$ .

$\frac{0}{0}$

Step 1.1.3.6.6

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.7

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to \frac{1}{3}} \frac{3 x^{2} + 5 x - 2}{6 x^{2} + x - 1} = lim_{x \to \frac{1}{3}} \frac{\frac{d}{d x} [3 x^{2} + 5 x - 2]}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \frac{1}{3}} \frac{\frac{d}{d x} [3 x^{2} + 5 x - 2]}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3.2

By the Sum Rule, the derivative of $3 x^{2} + 5 x - 2$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 2]$ .

$lim_{x \to \frac{1}{3}} \frac{\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 1.3.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to \frac{1}{3}} \frac{3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to \frac{1}{3}} \frac{3 (2 x) + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3.3.3

Multiply $2$ by $3$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 1.3.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5 \cdot 1 + \frac{d}{d x} [- 2]}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3.4.3

Multiply $5$ by $1$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5 + \frac{d}{d x} [- 2]}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5 + 0}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3.6

Add $6 x + 5$ and $0$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5}{\frac{d}{d x} [6 x^{2} + x - 1]}$

Step 1.3.7

By the Sum Rule, the derivative of $6 x^{2} + x - 1$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5}{\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 1.3.8

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 1.3.8.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5}{6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 1.3.8.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5}{6 (2 x) + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 1.3.8.3

Multiply $2$ by $6$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5}{12 x + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 1.3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5}{12 x + 1 + \frac{d}{d x} [- 1]}$

Step 1.3.10

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5}{12 x + 1 + 0}$

Step 1.3.11

Add $12 x + 1$ and $0$ .

$lim_{x \to \frac{1}{3}} \frac{6 x + 5}{12 x + 1}$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{1}{3}$ .

$\frac{lim_{x \to \frac{1}{3}} 6 x + 5}{lim_{x \to \frac{1}{3}} 12 x + 1}$

Step 2.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{3}$ .

$\frac{lim_{x \to \frac{1}{3}} 6 x + lim_{x \to \frac{1}{3}} 5}{lim_{x \to \frac{1}{3}} 12 x + 1}$

Step 2.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 lim_{x \to \frac{1}{3}} x + lim_{x \to \frac{1}{3}} 5}{lim_{x \to \frac{1}{3}} 12 x + 1}$

Step 2.4

Evaluate the limit of $5$ which is constant as $x$ approaches $\frac{1}{3}$ .

$\frac{6 lim_{x \to \frac{1}{3}} x + 5}{lim_{x \to \frac{1}{3}} 12 x + 1}$

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{3}$ .

$\frac{6 lim_{x \to \frac{1}{3}} x + 5}{lim_{x \to \frac{1}{3}} 12 x + lim_{x \to \frac{1}{3}} 1}$

Step 2.6

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$\frac{6 lim_{x \to \frac{1}{3}} x + 5}{12 lim_{x \to \frac{1}{3}} x + lim_{x \to \frac{1}{3}} 1}$

Step 2.7

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{1}{3}$ .

$\frac{6 lim_{x \to \frac{1}{3}} x + 5}{12 lim_{x \to \frac{1}{3}} x + 1}$

Step 3

Evaluate the limits by plugging in $\frac{1}{3}$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $\frac{1}{3}$ for $x$ .

$\frac{6 (\frac{1}{3}) + 5}{12 lim_{x \to \frac{1}{3}} x + 1}$

Step 3.2

Evaluate the limit of $x$ by plugging in $\frac{1}{3}$ for $x$ .

$\frac{6 (\frac{1}{3}) + 5}{12 (\frac{1}{3}) + 1}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1

Factor $3$ out of $6$ .

$\frac{3 (2) \frac{1}{3} + 5}{12 (\frac{1}{3}) + 1}$

Step 4.1.1.2

Cancel the common factor.

$\frac{3 \cdot 2 \frac{1}{3} + 5}{12 (\frac{1}{3}) + 1}$

Step 4.1.1.3

Rewrite the expression.

$\frac{2 + 5}{12 (\frac{1}{3}) + 1}$

Step 4.1.2

Add $2$ and $5$ .

$\frac{7}{12 (\frac{1}{3}) + 1}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Factor $3$ out of $12$ .

$\frac{7}{3 (4) \frac{1}{3} + 1}$

Step 4.2.1.2

Cancel the common factor.

$\frac{7}{3 \cdot 4 \frac{1}{3} + 1}$

Step 4.2.1.3

Rewrite the expression.

$\frac{7}{4 + 1}$

Step 4.2.2

Add $4$ and $1$ .

$\frac{7}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{5}$

Decimal Form:

$1.4$