Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.1.4

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

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

Step 1.1.2.1.5

Evaluate the limit of $x^{2}$ which is constant as $h$ approaches $0$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Combine the opposite terms in ${(x + 0)}^{2} - x^{2}$ .

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Step 1.1.2.3.1

Add $x$ and $0$ .

$\frac{x^{2} - x^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2

Subtract $x^{2}$ from $x^{2}$ .

$\frac{0}{lim_{h \to 0} h}$

Step 1.1.3

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

$\frac{0}{0}$

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.

$lim_{h \to 0} \frac{{(x + h)}^{2} - x^{2}}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [{(x + h)}^{2} - x^{2}]}{\frac{d}{d h} [h]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

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

Step 1.3.3

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 1.3.3.1

Apply the distributive property.

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

Step 1.3.3.2

Apply the distributive property.

$lim_{h \to 0} \frac{\frac{d}{d h} [x \cdot x + x h + h (x + h) - x^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.3.3

Apply the distributive property.

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

Step 1.3.4

Simplify and combine like terms.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

Multiply $x$ by $x$ .

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

Step 1.3.4.1.2

Multiply $h$ by $h$ .

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

Step 1.3.4.2

Add $x h$ and $h x$ .

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Step 1.3.4.2.1

Reorder $x$ and $h$ .

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

Step 1.3.4.2.2

Add $h x$ and $h x$ .

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

Step 1.3.5

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

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

Step 1.3.6

Since $x^{2}$ is constant with respect to $h$ , the derivative of $x^{2}$ with respect to $h$ is $0$ .

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

Step 1.3.7

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

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Step 1.3.7.1

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

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

Step 1.3.7.2

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

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

Step 1.3.7.3

Multiply $2$ by $1$ .

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

Step 1.3.8

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

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

Step 1.3.9

Since $- x^{2}$ is constant with respect to $h$ , the derivative of $- x^{2}$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{0 + 2 x + 2 h + 0}{\frac{d}{d h} [h]}$

Step 1.3.10

Simplify.

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Step 1.3.10.1

Combine terms.

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Step 1.3.10.1.1

Add $0$ and $2 x$ .

$lim_{h \to 0} \frac{2 x + 2 h + 0}{\frac{d}{d h} [h]}$

Step 1.3.10.1.2

Add $2 x + 2 h$ and $0$ .

$lim_{h \to 0} \frac{2 x + 2 h}{\frac{d}{d h} [h]}$

Step 1.3.10.2

Reorder terms.

$lim_{h \to 0} \frac{2 h + 2 x}{\frac{d}{d h} [h]}$

Step 1.3.11

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

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

Step 1.4

Divide $2 h + 2 x$ by $1$ .

$lim_{h \to 0} 2 h + 2 x$

Step 2

Evaluate the limit.

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Step 2.1

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

$lim_{h \to 0} 2 h + lim_{h \to 0} 2 x$

Step 2.2

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

$2 lim_{h \to 0} h + lim_{h \to 0} 2 x$

Step 2.3

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

$2 lim_{h \to 0} h + 2 x$

Step 3

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

$2 \cdot 0 + 2 x$

Step 4

Simplify the answer.

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Step 4.1

Multiply $2$ by $0$ .

$0 + 2 x$

Step 4.2

Add $0$ and $2 x$ .

$2 x$