Calculus Examples

$lim_{x \to 4} {(\frac{3 x - 4}{x^{2} + 3 x - 1})}^{\frac{1}{3}}$

Step 1

Move the exponent $\frac{1}{3}$ from ${(\frac{3 x - 4}{x^{2} + 3 x - 1})}^{\frac{1}{3}}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 4} \frac{3 x - 4}{x^{2} + 3 x - 1})}^{\frac{1}{3}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

${(\frac{lim_{x \to 4} 3 x - 4}{lim_{x \to 4} x^{2} + 3 x - 1})}^{\frac{1}{3}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

${(\frac{lim_{x \to 4} 3 x - lim_{x \to 4} 4}{lim_{x \to 4} x^{2} + 3 x - 1})}^{\frac{1}{3}}$

Step 4

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

${(\frac{3 lim_{x \to 4} x - lim_{x \to 4} 4}{lim_{x \to 4} x^{2} + 3 x - 1})}^{\frac{1}{3}}$

Step 5

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

${(\frac{3 lim_{x \to 4} x - 1 \cdot 4}{lim_{x \to 4} x^{2} + 3 x - 1})}^{\frac{1}{3}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

${(\frac{3 lim_{x \to 4} x - 1 \cdot 4}{lim_{x \to 4} x^{2} + lim_{x \to 4} 3 x - lim_{x \to 4} 1})}^{\frac{1}{3}}$

Step 7

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

${(\frac{3 lim_{x \to 4} x - 1 \cdot 4}{{(lim_{x \to 4} x)}^{2} + lim_{x \to 4} 3 x - lim_{x \to 4} 1})}^{\frac{1}{3}}$

Step 8

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

${(\frac{3 lim_{x \to 4} x - 1 \cdot 4}{{(lim_{x \to 4} x)}^{2} + 3 lim_{x \to 4} x - lim_{x \to 4} 1})}^{\frac{1}{3}}$

Step 9

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

${(\frac{3 lim_{x \to 4} x - 1 \cdot 4}{{(lim_{x \to 4} x)}^{2} + 3 lim_{x \to 4} x - 1 \cdot 1})}^{\frac{1}{3}}$

Step 10

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

${(\frac{3 \cdot 4 - 1 \cdot 4}{{(lim_{x \to 4} x)}^{2} + 3 lim_{x \to 4} x - 1 \cdot 1})}^{\frac{1}{3}}$

Step 10.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

${(\frac{3 \cdot 4 - 1 \cdot 4}{4^{2} + 3 lim_{x \to 4} x - 1 \cdot 1})}^{\frac{1}{3}}$

Step 10.3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

${(\frac{3 \cdot 4 - 1 \cdot 4}{4^{2} + 3 \cdot 4 - 1 \cdot 1})}^{\frac{1}{3}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $3$ by $4$ .

${(\frac{12 - 1 \cdot 4}{4^{2} + 3 \cdot 4 - 1 \cdot 1})}^{\frac{1}{3}}$

Step 11.1.2

Multiply $- 1$ by $4$ .

${(\frac{12 - 4}{4^{2} + 3 \cdot 4 - 1 \cdot 1})}^{\frac{1}{3}}$

Step 11.1.3

Subtract $4$ from $12$ .

${(\frac{8}{4^{2} + 3 \cdot 4 - 1 \cdot 1})}^{\frac{1}{3}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Raise $4$ to the power of $2$ .

${(\frac{8}{16 + 3 \cdot 4 - 1 \cdot 1})}^{\frac{1}{3}}$

Step 11.2.2

Multiply $3$ by $4$ .

${(\frac{8}{16 + 12 - 1 \cdot 1})}^{\frac{1}{3}}$

Step 11.2.3

Multiply $- 1$ by $1$ .

${(\frac{8}{16 + 12 - 1})}^{\frac{1}{3}}$

Step 11.2.4

Add $16$ and $12$ .

${(\frac{8}{28 - 1})}^{\frac{1}{3}}$

Step 11.2.5

Subtract $1$ from $28$ .

${(\frac{8}{27})}^{\frac{1}{3}}$

Step 11.3

Apply the product rule to $\frac{8}{27}$ .

$\frac{8^{\frac{1}{3}}}{27^{\frac{1}{3}}}$

Step 11.4

Simplify the numerator.

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Step 11.4.1

Rewrite $8$ as $2^{3}$ .

$\frac{{(2^{3})}^{\frac{1}{3}}}{27^{\frac{1}{3}}}$

Step 11.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2^{3 (\frac{1}{3})}}{27^{\frac{1}{3}}}$

Step 11.4.3

Cancel the common factor of $3$ .

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Step 11.4.3.1

Cancel the common factor.

$\frac{2^{3 (\frac{1}{3})}}{27^{\frac{1}{3}}}$

Step 11.4.3.2

Rewrite the expression.

$\frac{2^{1}}{27^{\frac{1}{3}}}$

Step 11.4.4

Evaluate the exponent.

$\frac{2}{27^{\frac{1}{3}}}$

Step 11.5

Simplify the denominator.

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Step 11.5.1

Rewrite $27$ as $3^{3}$ .

$\frac{2}{{(3^{3})}^{\frac{1}{3}}}$

Step 11.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2}{3^{3 (\frac{1}{3})}}$

Step 11.5.3

Cancel the common factor of $3$ .

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Step 11.5.3.1

Cancel the common factor.

$\frac{2}{3^{3 (\frac{1}{3})}}$

Step 11.5.3.2

Rewrite the expression.

$\frac{2}{3^{1}}$

Step 11.5.4

Evaluate the exponent.

$\frac{2}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$