Calculus Examples

$lim_{x \to 0} \frac{3 x + 2 x^{- 1}}{x + 4 x^{- 1}}$

Step 1

Simplify terms.

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Step 1.1

Simplify the limit argument.

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Step 1.1.1

Convert negative exponents to fractions.

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Step 1.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{3 x + 2 \frac{1}{x}}{x + 4 x^{- 1}}$

Step 1.1.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 0} \frac{3 x + 2 \frac{1}{x}}{x + 4 \frac{1}{x}}$

Step 1.1.2

Combine factors.

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Step 1.1.2.1

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

$lim_{x \to 0} \frac{3 x + \frac{2}{x}}{x + 4 \frac{1}{x}}$

Step 1.1.2.2

Combine $4$ and $\frac{1}{x}$ .

$lim_{x \to 0} \frac{3 x + \frac{2}{x}}{x + \frac{4}{x}}$

Step 1.1.3

Combine terms.

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Step 1.1.3.1

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to 0} \frac{\frac{3 x \cdot x}{x} + \frac{2}{x}}{x + \frac{4}{x}}$

Step 1.1.3.2

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{3 x \cdot x + 2}{x}}{x + \frac{4}{x}}$

Step 1.1.3.3

To write $x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to 0} \frac{\frac{3 x \cdot x + 2}{x}}{\frac{x \cdot x}{x} + \frac{4}{x}}$

Step 1.1.3.4

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{3 x \cdot x + 2}{x}}{\frac{x \cdot x + 4}{x}}$

Step 1.2

Simplify the limit argument.

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Step 1.2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{3 x \cdot x + 2}{x} \cdot \frac{x}{x \cdot x + 4}$

Step 1.2.2

Combine factors.

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Step 1.2.2.1

Raise $x$ to the power of $1$ .

$lim_{x \to 0} \frac{3 (x^{1} x) + 2}{x} \cdot \frac{x}{x \cdot x + 4}$

Step 1.2.2.2

Raise $x$ to the power of $1$ .

$lim_{x \to 0} \frac{3 (x^{1} x^{1}) + 2}{x} \cdot \frac{x}{x \cdot x + 4}$

Step 1.2.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 0} \frac{3 x^{1 + 1} + 2}{x} \cdot \frac{x}{x \cdot x + 4}$

Step 1.2.2.4

Add $1$ and $1$ .

$lim_{x \to 0} \frac{3 x^{2} + 2}{x} \cdot \frac{x}{x \cdot x + 4}$

Step 1.2.2.5

Raise $x$ to the power of $1$ .

$lim_{x \to 0} \frac{3 x^{2} + 2}{x} \cdot \frac{x}{x^{1} x + 4}$

Step 1.2.2.6

Raise $x$ to the power of $1$ .

$lim_{x \to 0} \frac{3 x^{2} + 2}{x} \cdot \frac{x}{x^{1} x^{1} + 4}$

Step 1.2.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 0} \frac{3 x^{2} + 2}{x} \cdot \frac{x}{x^{1 + 1} + 4}$

Step 1.2.2.8

Add $1$ and $1$ .

$lim_{x \to 0} \frac{3 x^{2} + 2}{x} \cdot \frac{x}{x^{2} + 4}$

Step 1.2.2.9

Multiply $\frac{3 x^{2} + 2}{x}$ by $\frac{x}{x^{2} + 4}$ .

$lim_{x \to 0} \frac{(3 x^{2} + 2) x}{x (x^{2} + 4)}$

Step 1.2.3

Cancel the common factor of $x$ .

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Step 1.2.3.1

Cancel the common factor.

$lim_{x \to 0} \frac{(3 x^{2} + 2) x}{x (x^{2} + 4)}$

Step 1.2.3.2

Rewrite the expression.

$lim_{x \to 0} \frac{3 x^{2} + 2}{x^{2} + 4}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 3 x^{2} + 2}{lim_{x \to 0} x^{2} + 4}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 3 x^{2} + lim_{x \to 0} 2}{lim_{x \to 0} x^{2} + 4}$

Step 4

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

$\frac{3 lim_{x \to 0} x^{2} + lim_{x \to 0} 2}{lim_{x \to 0} x^{2} + 4}$

Step 5

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

$\frac{3 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 2}{lim_{x \to 0} x^{2} + 4}$

Step 6

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{3 {(lim_{x \to 0} x)}^{2} + 2}{lim_{x \to 0} x^{2} + 4}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{3 {(lim_{x \to 0} x)}^{2} + 2}{lim_{x \to 0} x^{2} + lim_{x \to 0} 4}$

Step 8

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

$\frac{3 {(lim_{x \to 0} x)}^{2} + 2}{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 4}$

Step 9

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$\frac{3 {(lim_{x \to 0} x)}^{2} + 2}{{(lim_{x \to 0} x)}^{2} + 4}$

Step 10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{3 \cdot 0^{2} + 2}{{(lim_{x \to 0} x)}^{2} + 4}$

Step 10.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{3 \cdot 0^{2} + 2}{0^{2} + 4}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{3 \cdot 0 + 2}{0^{2} + 4}$

Step 11.1.2

Multiply $3$ by $0$ .

$\frac{0 + 2}{0^{2} + 4}$

Step 11.1.3

Add $0$ and $2$ .

$\frac{2}{0^{2} + 4}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{2}{0 + 4}$

Step 11.2.2

Add $0$ and $4$ .

$\frac{2}{4}$

Step 11.3

Cancel the common factor of $2$ and $4$ .

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Step 11.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 11.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 11.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$