Calculus Examples

$lim_{x \to 0} \frac{5 x^{3} + 8 x^{3}}{3 x^{4} - 16 x^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

Add $5 x^{3}$ and $8 x^{3}$ .

$lim_{x \to 0} \frac{13 x^{3}}{3 x^{4} - 16 x^{2}}$

Step 1.2

Move the term $13$ outside of the limit because it is constant with respect to $x$ .

$13 lim_{x \to 0} \frac{x^{3}}{3 x^{4} - 16 x^{2}}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$13 \frac{lim_{x \to 0} x^{3}}{lim_{x \to 0} 3 x^{4} - 16 x^{2}}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$13 \frac{{(lim_{x \to 0} x)}^{3}}{lim_{x \to 0} 3 x^{4} - 16 x^{2}}$

Step 2.1.2.2

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

$13 \frac{0^{3}}{lim_{x \to 0} 3 x^{4} - 16 x^{2}}$

Step 2.1.2.3

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

$13 \frac{0}{lim_{x \to 0} 3 x^{4} - 16 x^{2}}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

$13 \frac{0}{lim_{x \to 0} 3 x^{4} - lim_{x \to 0} 16 x^{2}}$

Step 2.1.3.2

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

$13 \frac{0}{3 lim_{x \to 0} x^{4} - lim_{x \to 0} 16 x^{2}}$

Step 2.1.3.3

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$13 \frac{0}{3 {(lim_{x \to 0} x)}^{4} - lim_{x \to 0} 16 x^{2}}$

Step 2.1.3.4

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$13 \frac{0}{3 {(lim_{x \to 0} x)}^{4} - 16 lim_{x \to 0} x^{2}}$

Step 2.1.3.5

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

$13 \frac{0}{3 {(lim_{x \to 0} x)}^{4} - 16 {(lim_{x \to 0} x)}^{2}}$

Step 2.1.3.6

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

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Step 2.1.3.6.1

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

$13 \frac{0}{3 \cdot 0^{4} - 16 {(lim_{x \to 0} x)}^{2}}$

Step 2.1.3.6.2

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

$13 \frac{0}{3 \cdot 0^{4} - 16 \cdot 0^{2}}$

Step 2.1.3.7

Simplify the answer.

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Step 2.1.3.7.1

Simplify each term.

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Step 2.1.3.7.1.1

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

$13 \frac{0}{3 \cdot 0 - 16 \cdot 0^{2}}$

Step 2.1.3.7.1.2

Multiply $3$ by $0$ .

$13 \frac{0}{0 - 16 \cdot 0^{2}}$

Step 2.1.3.7.1.3

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

$13 \frac{0}{0 - 16 \cdot 0}$

Step 2.1.3.7.1.4

Multiply $- 16$ by $0$ .

$13 \frac{0}{0 + 0}$

Step 2.1.3.7.2

Add $0$ and $0$ .

$13 (\frac{0}{0})$

Step 2.1.3.7.3

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

Undefined

$13 (\frac{0}{0})$

Step 2.1.3.8

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

Undefined

$13 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$13 (\frac{0}{0})$

Step 2.2

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.

$lim_{x \to 0} \frac{x^{3}}{3 x^{4} - 16 x^{2}} = lim_{x \to 0} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [3 x^{4} - 16 x^{2}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$13 lim_{x \to 0} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [3 x^{4} - 16 x^{2}]}$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$13 lim_{x \to 0} \frac{3 x^{2}}{\frac{d}{d x} [3 x^{4} - 16 x^{2}]}$

Step 2.3.3

By the Sum Rule, the derivative of $3 x^{4} - 16 x^{2}$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 16 x^{2}]$ .

$13 lim_{x \to 0} \frac{3 x^{2}}{\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 16 x^{2}]}$

Step 2.3.4

Evaluate $\frac{d}{d x} [3 x^{4}]$ .

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Step 2.3.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$13 lim_{x \to 0} \frac{3 x^{2}}{3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16 x^{2}]}$

Step 2.3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$13 lim_{x \to 0} \frac{3 x^{2}}{3 \cdot 4 x^{3} + \frac{d}{d x} [- 16 x^{2}]}$

Step 2.3.4.3

Multiply $4$ by $3$ .

$13 lim_{x \to 0} \frac{3 x^{2}}{12 x^{3} + \frac{d}{d x} [- 16 x^{2}]}$

Step 2.3.5

Evaluate $\frac{d}{d x} [- 16 x^{2}]$ .

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Step 2.3.5.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x^{2}$ with respect to $x$ is $- 16 \frac{d}{d x} [x^{2}]$ .

$13 lim_{x \to 0} \frac{3 x^{2}}{12 x^{3} - 16 \frac{d}{d x} [x^{2}]}$

Step 2.3.5.2

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

$13 lim_{x \to 0} \frac{3 x^{2}}{12 x^{3} - 16 \cdot 2 x}$

Step 2.3.5.3

Multiply $2$ by $- 16$ .

$13 lim_{x \to 0} \frac{3 x^{2}}{12 x^{3} - 32 x}$

Step 3

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

$13 \cdot 3 lim_{x \to 0} \frac{x^{2}}{12 x^{3} - 32 x}$

Step 4

Apply L'Hospital's rule.

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Step 4.1

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

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Step 4.1.1

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

$13 \cdot 3 \frac{lim_{x \to 0} x^{2}}{lim_{x \to 0} 12 x^{3} - 32 x}$

Step 4.1.2

Evaluate the limit of the numerator.

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Step 4.1.2.1

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

$13 \cdot 3 \frac{{(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} 12 x^{3} - 32 x}$

Step 4.1.2.2

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

$13 \cdot 3 \frac{0^{2}}{lim_{x \to 0} 12 x^{3} - 32 x}$

Step 4.1.2.3

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

$13 \cdot 3 \frac{0}{lim_{x \to 0} 12 x^{3} - 32 x}$

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

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

$13 \cdot 3 \frac{0}{lim_{x \to 0} 12 x^{3} - lim_{x \to 0} 32 x}$

Step 4.1.3.2

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

$13 \cdot 3 \frac{0}{12 lim_{x \to 0} x^{3} - lim_{x \to 0} 32 x}$

Step 4.1.3.3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$13 \cdot 3 \frac{0}{12 {(lim_{x \to 0} x)}^{3} - lim_{x \to 0} 32 x}$

Step 4.1.3.4

Move the term $32$ outside of the limit because it is constant with respect to $x$ .

$13 \cdot 3 \frac{0}{12 {(lim_{x \to 0} x)}^{3} - 32 lim_{x \to 0} x}$

Step 4.1.3.5

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

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Step 4.1.3.5.1

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

$13 \cdot 3 \frac{0}{12 \cdot 0^{3} - 32 lim_{x \to 0} x}$

Step 4.1.3.5.2

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

$13 \cdot 3 \frac{0}{12 \cdot 0^{3} - 32 \cdot 0}$

Step 4.1.3.6

Simplify the answer.

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Step 4.1.3.6.1

Simplify each term.

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Step 4.1.3.6.1.1

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

$13 \cdot 3 \frac{0}{12 \cdot 0 - 32 \cdot 0}$

Step 4.1.3.6.1.2

Multiply $12$ by $0$ .

$13 \cdot 3 \frac{0}{0 - 32 \cdot 0}$

Step 4.1.3.6.1.3

Multiply $- 32$ by $0$ .

$13 \cdot 3 \frac{0}{0 + 0}$

Step 4.1.3.6.2

Add $0$ and $0$ .

$13 \cdot 3 \frac{0}{0}$

Step 4.1.3.6.3

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

Undefined

$13 \cdot 3 \frac{0}{0}$

Step 4.1.3.7

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

Undefined

$13 \cdot 3 \frac{0}{0}$

Step 4.1.4

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

Undefined

$13 \cdot 3 \frac{0}{0}$

Step 4.2

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.

$lim_{x \to 0} \frac{x^{2}}{12 x^{3} - 32 x} = lim_{x \to 0} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [12 x^{3} - 32 x]}$

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$13 \cdot 3 lim_{x \to 0} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [12 x^{3} - 32 x]}$

Step 4.3.2

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

$13 \cdot 3 lim_{x \to 0} \frac{2 x}{\frac{d}{d x} [12 x^{3} - 32 x]}$

Step 4.3.3

By the Sum Rule, the derivative of $12 x^{3} - 32 x$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [- 32 x]$ .

$13 \cdot 3 lim_{x \to 0} \frac{2 x}{\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [- 32 x]}$

Step 4.3.4

Evaluate $\frac{d}{d x} [12 x^{3}]$ .

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Step 4.3.4.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{3}$ with respect to $x$ is $12 \frac{d}{d x} [x^{3}]$ .

$13 \cdot 3 lim_{x \to 0} \frac{2 x}{12 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 32 x]}$

Step 4.3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$13 \cdot 3 lim_{x \to 0} \frac{2 x}{12 \cdot 3 x^{2} + \frac{d}{d x} [- 32 x]}$

Step 4.3.4.3

Multiply $3$ by $12$ .

$13 \cdot 3 lim_{x \to 0} \frac{2 x}{36 x^{2} + \frac{d}{d x} [- 32 x]}$

Step 4.3.5

Evaluate $\frac{d}{d x} [- 32 x]$ .

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Step 4.3.5.1

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32 x$ with respect to $x$ is $- 32 \frac{d}{d x} [x]$ .

$13 \cdot 3 lim_{x \to 0} \frac{2 x}{36 x^{2} - 32 \frac{d}{d x} [x]}$

Step 4.3.5.2

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

$13 \cdot 3 lim_{x \to 0} \frac{2 x}{36 x^{2} - 32 \cdot 1}$

Step 4.3.5.3

Multiply $- 32$ by $1$ .

$13 \cdot 3 lim_{x \to 0} \frac{2 x}{36 x^{2} - 32}$

Step 4.4

Cancel the common factor of $2$ and $36 x^{2} - 32$ .

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Step 4.4.1

Factor $2$ out of $2 x$ .

$13 \cdot 3 lim_{x \to 0} \frac{2 (x)}{36 x^{2} - 32}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $36 x^{2}$ .

$13 \cdot 3 lim_{x \to 0} \frac{2 (x)}{2 \cdot 18 x^{2} - 32}$

Step 4.4.2.2

Factor $2$ out of $- 32$ .

$13 \cdot 3 lim_{x \to 0} \frac{2 x}{2 \cdot 18 x^{2} + 2 \cdot - 16}$

Step 4.4.2.3

Factor $2$ out of $2 (18 x^{2}) + 2 (- 16)$ .

$13 \cdot 3 lim_{x \to 0} \frac{2 (x)}{2 (18 x^{2} - 16)}$

Step 4.4.2.4

Cancel the common factor.

$13 \cdot 3 lim_{x \to 0} \frac{2 x}{2 (18 x^{2} - 16)}$

Step 4.4.2.5

Rewrite the expression.

$13 \cdot 3 lim_{x \to 0} \frac{x}{18 x^{2} - 16}$

Step 5

Evaluate the limit.

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Step 5.1

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

$13 \cdot 3 \frac{lim_{x \to 0} x}{lim_{x \to 0} 18 x^{2} - 16}$

Step 5.2

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

$13 \cdot 3 \frac{lim_{x \to 0} x}{lim_{x \to 0} 18 x^{2} - lim_{x \to 0} 16}$

Step 5.3

Move the term $18$ outside of the limit because it is constant with respect to $x$ .

$13 \cdot 3 \frac{lim_{x \to 0} x}{18 lim_{x \to 0} x^{2} - lim_{x \to 0} 16}$

Step 5.4

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

$13 \cdot 3 \frac{lim_{x \to 0} x}{18 {(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 16}$

Step 5.5

Evaluate the limit of $16$ which is constant as $x$ approaches $0$ .

$13 \cdot 3 \frac{lim_{x \to 0} x}{18 {(lim_{x \to 0} x)}^{2} - 1 \cdot 16}$

Step 6

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

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Step 6.1

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

$13 \cdot 3 \frac{0}{18 {(lim_{x \to 0} x)}^{2} - 1 \cdot 16}$

Step 6.2

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

$13 \cdot 3 \frac{0}{18 \cdot 0^{2} - 1 \cdot 16}$

Step 7

Simplify the answer.

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Step 7.1

Multiply $13$ by $3$ .

$39 \frac{0}{18 \cdot 0^{2} - 1 \cdot 16}$

Step 7.2

Cancel the common factor of $0$ and $18 \cdot 0^{2} - 1 \cdot 16$ .

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Step 7.2.1

Factor $- 1$ out of $18 \cdot 0^{2}$ .

$39 \frac{0}{- (- 18 \cdot 0^{2}) - 1 \cdot 16}$

Step 7.2.2

Factor $- 1$ out of $- 1 \cdot 16$ .

$39 \frac{0}{- (- 18 \cdot 0^{2}) - 1 (16)}$

Step 7.2.3

Factor $- 1$ out of $- (- 18 \cdot 0^{2}) - 1 (16)$ .

$39 \frac{0}{- (- 18 \cdot 0^{2} + 16)}$

Step 7.2.4

Rewrite $- (- 18 \cdot 0^{2} + 16)$ as $- 1 (- 18 \cdot 0^{2} + 16)$ .

$39 \frac{0}{- 1 (- 18 \cdot 0^{2} + 16)}$

Step 7.2.5

Factor $2$ out of $0$ .

$39 \frac{2 (0)}{- 1 (- 18 \cdot 0^{2} + 16)}$

Step 7.2.6

Cancel the common factors.

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Step 7.2.6.1

Factor $2$ out of $- 1 (- 18 \cdot 0^{2} + 16)$ .

$39 \frac{2 (0)}{2 (- 1 (- 9 \cdot 0^{2} + 8))}$

Step 7.2.6.2

Cancel the common factor.

$39 \frac{2 \cdot 0}{2 (- 1 (- 9 \cdot 0^{2} + 8))}$

Step 7.2.6.3

Rewrite the expression.

$39 \frac{0}{- 1 (- 9 \cdot 0^{2} + 8)}$

Step 7.3

Simplify the denominator.

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Step 7.3.1

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

$39 \frac{0}{- 1 (- 9 \cdot 0 + 8)}$

Step 7.3.2

Multiply $- 9$ by $0$ .

$39 \frac{0}{- 1 (0 + 8)}$

Step 7.3.3

Add $0$ and $8$ .

$39 \frac{0}{- 1 \cdot 8}$

Step 7.4

Multiply $- 1$ by $8$ .

$39 (\frac{0}{- 8})$

Step 7.5

Divide $0$ by $- 8$ .

$39 \cdot 0$

Step 7.6

Multiply $39$ by $0$ .

$0$