Calculus Examples

$lim_{x \to 0^{+}} \frac{3 (e^{x} - 1 - x)}{10 x^{3}}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to 0^{+}} 3 (e^{x} - 1 - x)}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

$\frac{3 lim_{x \to 0^{+}} e^{x} - 1 - x}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2.2

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

$\frac{3 (lim_{x \to 0^{+}} e^{x} - lim_{x \to 0^{+}} 1 - lim_{x \to 0^{+}} x)}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2.3

Move the limit into the exponent.

$\frac{3 (e^{lim_{x \to 0^{+}} x} - lim_{x \to 0^{+}} 1 - lim_{x \to 0^{+}} x)}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{3 (e^{lim_{x \to 0^{+}} x} - 1 \cdot 1 - lim_{x \to 0^{+}} x)}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2.5

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

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Step 1.2.5.1

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

$\frac{3 (e^{0} - 1 \cdot 1 - lim_{x \to 0^{+}} x)}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2.5.2

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

$\frac{3 (e^{0} - 1 \cdot 1 + 0)}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2.6

Simplify the answer.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

Anything raised to $0$ is $1$ .

$\frac{3 (1 - 1 \cdot 1 + 0)}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2.6.1.2

Multiply $- 1$ by $1$ .

$\frac{3 (1 - 1 + 0)}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2.6.2

Subtract $1$ from $1$ .

$\frac{3 (0 + 0)}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2.6.3

Add $0$ and $0$ .

$\frac{3 \cdot 0}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.2.6.4

Multiply $3$ by $0$ .

$\frac{0}{lim_{x \to 0^{+}} 10 x^{3}}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{10 lim_{x \to 0^{+}} x^{3}}$

Step 1.3.1.2

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

$\frac{0}{10 {(lim_{x \to 0^{+}} x)}^{3}}$

Step 1.3.2

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

$\frac{0}{10 \cdot 0^{3}}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

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

$\frac{0}{10 \cdot 0}$

Step 1.3.3.2

Multiply $10$ by $0$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 0^{+}} \frac{3 (e^{x} - 1 - x)}{10 x^{3}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [3 (e^{x} - 1 - x)]}{\frac{d}{d x} [10 x^{3}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0^{+}} \frac{\frac{d}{d x} [3 (e^{x} - 1 - x)]}{\frac{d}{d x} [10 x^{3}]}$

Step 3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 (e^{x} - 1 - x)$ with respect to $x$ is $3 \frac{d}{d x} [e^{x} - 1 - x]$ .

$lim_{x \to 0^{+}} \frac{3 \frac{d}{d x} [e^{x} - 1 - x]}{\frac{d}{d x} [10 x^{3}]}$

Step 3.3

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

$lim_{x \to 0^{+}} \frac{3 (\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- x])}{\frac{d}{d x} [10 x^{3}]}$

Step 3.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$lim_{x \to 0^{+}} \frac{3 (e^{x} + \frac{d}{d x} [- 1] + \frac{d}{d x} [- x])}{\frac{d}{d x} [10 x^{3}]}$

Step 3.5

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

$lim_{x \to 0^{+}} \frac{3 (e^{x} + 0 + \frac{d}{d x} [- x])}{\frac{d}{d x} [10 x^{3}]}$

Step 3.6

Add $e^{x}$ and $0$ .

$lim_{x \to 0^{+}} \frac{3 (e^{x} + \frac{d}{d x} [- x])}{\frac{d}{d x} [10 x^{3}]}$

Step 3.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$lim_{x \to 0^{+}} \frac{3 (e^{x} - \frac{d}{d x} [x])}{\frac{d}{d x} [10 x^{3}]}$

Step 3.8

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

$lim_{x \to 0^{+}} \frac{3 (e^{x} - 1 \cdot 1)}{\frac{d}{d x} [10 x^{3}]}$

Step 3.9

Multiply $- 1$ by $1$ .

$lim_{x \to 0^{+}} \frac{3 (e^{x} - 1)}{\frac{d}{d x} [10 x^{3}]}$

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

$lim_{x \to 0^{+}} \frac{3 e^{x} + 3 \cdot - 1}{\frac{d}{d x} [10 x^{3}]}$

Step 3.10.2

Multiply $3$ by $- 1$ .

$lim_{x \to 0^{+}} \frac{3 e^{x} - 3}{\frac{d}{d x} [10 x^{3}]}$

Step 3.11

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{3}$ with respect to $x$ is $10 \frac{d}{d x} [x^{3}]$ .

$lim_{x \to 0^{+}} \frac{3 e^{x} - 3}{10 \frac{d}{d x} [x^{3}]}$

Step 3.12

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

$lim_{x \to 0^{+}} \frac{3 e^{x} - 3}{10 (3 x^{2})}$

Step 3.13

Multiply $3$ by $10$ .

$lim_{x \to 0^{+}} \frac{3 e^{x} - 3}{30 x^{2}}$

Step 4

Cancel the common factor of $3 e^{x} - 3$ and $30$ .

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Step 4.1

Factor $3$ out of $3 e^{x}$ .

$lim_{x \to 0^{+}} \frac{3 (e^{x}) - 3}{30 x^{2}}$

Step 4.2

Factor $3$ out of $- 3$ .

$lim_{x \to 0^{+}} \frac{3 (e^{x}) + 3 (- 1)}{30 x^{2}}$

Step 4.3

Factor $3$ out of $3 (e^{x}) + 3 (- 1)$ .

$lim_{x \to 0^{+}} \frac{3 (e^{x} - 1)}{30 x^{2}}$

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $3$ out of $30 x^{2}$ .

$lim_{x \to 0^{+}} \frac{3 (e^{x} - 1)}{3 (10 x^{2})}$

Step 4.4.2

Cancel the common factor.

$lim_{x \to 0^{+}} \frac{3 (e^{x} - 1)}{3 (10 x^{2})}$

Step 4.4.3

Rewrite the expression.

$lim_{x \to 0^{+}} \frac{e^{x} - 1}{10 x^{2}}$

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

$\frac{lim_{x \to 0^{+}} e^{x} - 1}{lim_{x \to 0^{+}} 10 x^{2}}$

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

Evaluate the limit.

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Step 5.1.2.1.1

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

$\frac{lim_{x \to 0^{+}} e^{x} - lim_{x \to 0^{+}} 1}{lim_{x \to 0^{+}} 10 x^{2}}$

Step 5.1.2.1.2

Move the limit into the exponent.

$\frac{e^{lim_{x \to 0^{+}} x} - lim_{x \to 0^{+}} 1}{lim_{x \to 0^{+}} 10 x^{2}}$

Step 5.1.2.1.3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{e^{lim_{x \to 0^{+}} x} - 1 \cdot 1}{lim_{x \to 0^{+}} 10 x^{2}}$

Step 5.1.2.2

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

$\frac{e^{0} - 1 \cdot 1}{lim_{x \to 0^{+}} 10 x^{2}}$

Step 5.1.2.3

Simplify the answer.

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Step 5.1.2.3.1

Simplify each term.

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Step 5.1.2.3.1.1

Anything raised to $0$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to 0^{+}} 10 x^{2}}$

Step 5.1.2.3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 0^{+}} 10 x^{2}}$

Step 5.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 0^{+}} 10 x^{2}}$

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

Evaluate the limit.

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Step 5.1.3.1.1

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{10 lim_{x \to 0^{+}} x^{2}}$

Step 5.1.3.1.2

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

$\frac{0}{10 {(lim_{x \to 0^{+}} x)}^{2}}$

Step 5.1.3.2

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

$\frac{0}{10 \cdot 0^{2}}$

Step 5.1.3.3

Simplify the answer.

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Step 5.1.3.3.1

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

$\frac{0}{10 \cdot 0}$

Step 5.1.3.3.2

Multiply $10$ by $0$ .

$\frac{0}{0}$

Step 5.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 5.1.3.4

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

Undefined

$\frac{0}{0}$

Step 5.1.4

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

Undefined

$\frac{0}{0}$

Step 5.2

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.

$lim_{x \to 0^{+}} \frac{e^{x} - 1}{10 x^{2}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [e^{x} - 1]}{\frac{d}{d x} [10 x^{2}]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0^{+}} \frac{\frac{d}{d x} [e^{x} - 1]}{\frac{d}{d x} [10 x^{2}]}$

Step 5.3.2

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

$lim_{x \to 0^{+}} \frac{\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 1]}{\frac{d}{d x} [10 x^{2}]}$

Step 5.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$lim_{x \to 0^{+}} \frac{e^{x} + \frac{d}{d x} [- 1]}{\frac{d}{d x} [10 x^{2}]}$

Step 5.3.4

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

$lim_{x \to 0^{+}} \frac{e^{x} + 0}{\frac{d}{d x} [10 x^{2}]}$

Step 5.3.5

Add $e^{x}$ and $0$ .

$lim_{x \to 0^{+}} \frac{e^{x}}{\frac{d}{d x} [10 x^{2}]}$

Step 5.3.6

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{2}$ with respect to $x$ is $10 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 0^{+}} \frac{e^{x}}{10 \frac{d}{d x} [x^{2}]}$

Step 5.3.7

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

$lim_{x \to 0^{+}} \frac{e^{x}}{10 (2 x)}$

Step 5.3.8

Multiply $2$ by $10$ .

$lim_{x \to 0^{+}} \frac{e^{x}}{20 x}$

Step 6

Since the numerator is positive and the denominator $20 x$ approaches zero and is greater than zero for $x$ near $0$ to the right, the function increases without bound.

$\infty$