Calculus Examples

$lim_{x \to 0} \frac{2 x^{3} + 5 x^{2} - 10 x}{x^{2} + 2 x}$

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.

$\frac{lim_{x \to 0} 2 x^{3} + 5 x^{2} - 10 x}{lim_{x \to 0} x^{2} + 2 x}$

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 $0$ .

$\frac{lim_{x \to 0} 2 x^{3} + lim_{x \to 0} 5 x^{2} - lim_{x \to 0} 10 x}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 0} x^{3} + lim_{x \to 0} 5 x^{2} - lim_{x \to 0} 10 x}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.3

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

$\frac{2 {(lim_{x \to 0} x)}^{3} + lim_{x \to 0} 5 x^{2} - lim_{x \to 0} 10 x}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.4

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

$\frac{2 {(lim_{x \to 0} x)}^{3} + 5 lim_{x \to 0} x^{2} - lim_{x \to 0} 10 x}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.5

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

$\frac{2 {(lim_{x \to 0} x)}^{3} + 5 {(lim_{x \to 0} x)}^{2} - lim_{x \to 0} 10 x}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.6

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

$\frac{2 {(lim_{x \to 0} x)}^{3} + 5 {(lim_{x \to 0} x)}^{2} - 10 lim_{x \to 0} x}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.7

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

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Step 1.1.2.7.1

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

$\frac{2 \cdot 0^{3} + 5 {(lim_{x \to 0} x)}^{2} - 10 lim_{x \to 0} x}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.7.2

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

$\frac{2 \cdot 0^{3} + 5 \cdot 0^{2} - 10 lim_{x \to 0} x}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.7.3

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

$\frac{2 \cdot 0^{3} + 5 \cdot 0^{2} - 10 \cdot 0}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.8

Simplify the answer.

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Step 1.1.2.8.1

Simplify each term.

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Step 1.1.2.8.1.1

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

$\frac{2 \cdot 0 + 5 \cdot 0^{2} - 10 \cdot 0}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.8.1.2

Multiply $2$ by $0$ .

$\frac{0 + 5 \cdot 0^{2} - 10 \cdot 0}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.8.1.3

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

$\frac{0 + 5 \cdot 0 - 10 \cdot 0}{lim_{x \to 0} x^{2} + 2 x}$

Step 1.1.2.8.1.4

Multiply $5$ by $0$ .

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

Step 1.1.2.8.1.5

Multiply $- 10$ by $0$ .

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

Step 1.1.2.8.2

Add $0$ and $0$ .

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

Step 1.1.2.8.3

Add $0$ and $0$ .

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

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 $0$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.1.3.4

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

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Step 1.1.3.4.1

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

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

Step 1.1.3.4.2

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

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

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Simplify each term.

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Step 1.1.3.5.1.1

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

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

Step 1.1.3.5.1.2

Multiply $2$ by $0$ .

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

Step 1.1.3.5.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.1.3.5.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6

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.

$lim_{x \to 0} \frac{2 x^{3} + 5 x^{2} - 10 x}{x^{2} + 2 x} = lim_{x \to 0} \frac{\frac{d}{d x} [2 x^{3} + 5 x^{2} - 10 x]}{\frac{d}{d x} [x^{2} + 2 x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [2 x^{3} + 5 x^{2} - 10 x]}{\frac{d}{d x} [x^{2} + 2 x]}$

Step 1.3.2

By the Sum Rule, the derivative of $2 x^{3} + 5 x^{2} - 10 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 10 x]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 10 x]}{\frac{d}{d x} [x^{2} + 2 x]}$

Step 1.3.3

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

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Step 1.3.3.1

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

$lim_{x \to 0} \frac{2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 10 x]}{\frac{d}{d x} [x^{2} + 2 x]}$

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 $n = 3$ .

$lim_{x \to 0} \frac{2 (3 x^{2}) + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 10 x]}{\frac{d}{d x} [x^{2} + 2 x]}$

Step 1.3.3.3

Multiply $3$ by $2$ .

$lim_{x \to 0} \frac{6 x^{2} + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 10 x]}{\frac{d}{d x} [x^{2} + 2 x]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 1.3.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 0} \frac{6 x^{2} + 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x]}{\frac{d}{d x} [x^{2} + 2 x]}$

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 $n = 2$ .

$lim_{x \to 0} \frac{6 x^{2} + 5 (2 x) + \frac{d}{d x} [- 10 x]}{\frac{d}{d x} [x^{2} + 2 x]}$

Step 1.3.4.3

Multiply $2$ by $5$ .

$lim_{x \to 0} \frac{6 x^{2} + 10 x + \frac{d}{d x} [- 10 x]}{\frac{d}{d x} [x^{2} + 2 x]}$

Step 1.3.5

Evaluate $\frac{d}{d x} [- 10 x]$ .

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Step 1.3.5.1

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

$lim_{x \to 0} \frac{6 x^{2} + 10 x - 10 \frac{d}{d x} [x]}{\frac{d}{d x} [x^{2} + 2 x]}$

Step 1.3.5.2

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

$lim_{x \to 0} \frac{6 x^{2} + 10 x - 10 \cdot 1}{\frac{d}{d x} [x^{2} + 2 x]}$

Step 1.3.5.3

Multiply $- 10$ by $1$ .

$lim_{x \to 0} \frac{6 x^{2} + 10 x - 10}{\frac{d}{d x} [x^{2} + 2 x]}$

Step 1.3.6

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

$lim_{x \to 0} \frac{6 x^{2} + 10 x - 10}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]}$

Step 1.3.7

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

$lim_{x \to 0} \frac{6 x^{2} + 10 x - 10}{2 x + \frac{d}{d x} [2 x]}$

Step 1.3.8

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

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Step 1.3.8.1

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

$lim_{x \to 0} \frac{6 x^{2} + 10 x - 10}{2 x + 2 \frac{d}{d x} [x]}$

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 $n = 1$ .

$lim_{x \to 0} \frac{6 x^{2} + 10 x - 10}{2 x + 2 \cdot 1}$

Step 1.3.8.3

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{6 x^{2} + 10 x - 10}{2 x + 2}$

Step 1.4

Cancel the common factor of $6 x^{2} + 10 x - 10$ and $2 x + 2$ .

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Step 1.4.1

Factor $2$ out of $6 x^{2}$ .

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

Step 1.4.2

Factor $2$ out of $10 x$ .

$lim_{x \to 0} \frac{2 (3 x^{2}) + 2 (5 x) - 10}{2 x + 2}$

Step 1.4.3

Factor $2$ out of $2 (3 x^{2}) + 2 (5 x)$ .

$lim_{x \to 0} \frac{2 (3 x^{2} + 5 x) - 10}{2 x + 2}$

Step 1.4.4

Factor $2$ out of $- 10$ .

$lim_{x \to 0} \frac{2 (3 x^{2} + 5 x) + 2 \cdot - 5}{2 x + 2}$

Step 1.4.5

Factor $2$ out of $2 (3 x^{2} + 5 x) + 2 (- 5)$ .

$lim_{x \to 0} \frac{2 (3 x^{2} + 5 x - 5)}{2 x + 2}$

Step 1.4.6

Cancel the common factors.

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Step 1.4.6.1

Factor $2$ out of $2 x$ .

$lim_{x \to 0} \frac{2 (3 x^{2} + 5 x - 5)}{2 (x) + 2}$

Step 1.4.6.2

Factor $2$ out of $2$ .

$lim_{x \to 0} \frac{2 (3 x^{2} + 5 x - 5)}{2 (x) + 2 (1)}$

Step 1.4.6.3

Factor $2$ out of $2 (x) + 2 (1)$ .

$lim_{x \to 0} \frac{2 (3 x^{2} + 5 x - 5)}{2 (x + 1)}$

Step 1.4.6.4

Cancel the common factor.

$lim_{x \to 0} \frac{2 (3 x^{2} + 5 x - 5)}{2 (x + 1)}$

Step 1.4.6.5

Rewrite the expression.

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

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 $0$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$\frac{3 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 5 x - lim_{x \to 0} 5}{lim_{x \to 0} x + 1}$

Step 2.5

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

$\frac{3 {(lim_{x \to 0} x)}^{2} + 5 lim_{x \to 0} x - lim_{x \to 0} 5}{lim_{x \to 0} x + 1}$

Step 2.6

Evaluate the limit of $5$ which is constant as $x$ approaches $0$ .

$\frac{3 {(lim_{x \to 0} x)}^{2} + 5 lim_{x \to 0} x - 1 \cdot 5}{lim_{x \to 0} x + 1}$

Step 2.7

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

$\frac{3 {(lim_{x \to 0} x)}^{2} + 5 lim_{x \to 0} x - 1 \cdot 5}{lim_{x \to 0} x + lim_{x \to 0} 1}$

Step 2.8

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

$\frac{3 {(lim_{x \to 0} x)}^{2} + 5 lim_{x \to 0} x - 1 \cdot 5}{lim_{x \to 0} x + 1}$

Step 3

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

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Step 3.1

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

$\frac{3 \cdot 0^{2} + 5 lim_{x \to 0} x - 1 \cdot 5}{lim_{x \to 0} x + 1}$

Step 3.2

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

$\frac{3 \cdot 0^{2} + 5 \cdot 0 - 1 \cdot 5}{lim_{x \to 0} x + 1}$

Step 3.3

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

$\frac{3 \cdot 0^{2} + 5 \cdot 0 - 1 \cdot 5}{0 + 1}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$\frac{3 \cdot 0 + 5 \cdot 0 - 1 \cdot 5}{0 + 1}$

Step 4.1.2

Multiply $3$ by $0$ .

$\frac{0 + 5 \cdot 0 - 1 \cdot 5}{0 + 1}$

Step 4.1.3

Multiply $5$ by $0$ .

$\frac{0 + 0 - 1 \cdot 5}{0 + 1}$

Step 4.1.4

Multiply $- 1$ by $5$ .

$\frac{0 + 0 - 5}{0 + 1}$

Step 4.1.5

Add $0$ and $0$ .

$\frac{0 - 5}{0 + 1}$

Step 4.1.6

Subtract $5$ from $0$ .

$\frac{- 5}{0 + 1}$

Step 4.2

Add $0$ and $1$ .

$\frac{- 5}{1}$

Step 4.3

Divide $- 5$ by $1$ .

$- 5$