Calculus Examples

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

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.

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

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 $h$ approaches $0$ .

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

Step 1.1.2.2

Add $x$ and $0$ .

$\frac{x - lim_{h \to 0} \arctan (x)}{lim_{h \to 0} h}$

Step 1.1.2.3

Substitute $t$ for $x + h$ and let $t$ approach $x$ since $lim_{h \to 0} x + h = x$ .

$\frac{lim_{t \to x} \arctan (t) - lim_{h \to 0} \arctan (x)}{lim_{h \to 0} h}$

Step 1.1.2.4

Evaluate the limit of $\arctan (x)$ which is constant as $h$ approaches $0$ .

$\frac{lim_{t \to x} \arctan (t) - \arctan (x)}{lim_{h \to 0} h}$

Step 1.1.2.5

Add $x$ and $0$ .

$\frac{\arctan (x) - \arctan (x)}{lim_{h \to 0} h}$

Step 1.1.2.6

Subtract $\arctan (x)$ from $\arctan (x)$ .

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

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

Evaluate $\frac{d}{d h} [\arctan (x + h)]$ .

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Step 1.3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = \arctan (h)$ and $g (h) = x + h$ .

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Step 1.3.3.1.1

To apply the Chain Rule, set $u$ as $x + h$ .

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

Step 1.3.3.1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

$lim_{h \to 0} \frac{\frac{1}{1 + u^{2}} \frac{d}{d h} [x + h] + \frac{d}{d h} [- \arctan (x)]}{\frac{d}{d h} [h]}$

Step 1.3.3.1.3

Replace all occurrences of $u$ with $x + h$ .

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

Step 1.3.3.2

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

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

Step 1.3.3.3

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

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

Step 1.3.3.4

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

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

Step 1.3.3.5

Add $0$ and $1$ .

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

Step 1.3.3.6

Multiply $\frac{1}{1 + {(x + h)}^{2}}$ by $1$ .

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

Step 1.3.4

Since $- \arctan (x)$ is constant with respect to $h$ , the derivative of $- \arctan (x)$ with respect to $h$ is $0$ .

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

Step 1.3.5

Simplify.

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Step 1.3.5.1

Add $\frac{1}{1 + {(x + h)}^{2}}$ and $0$ .

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

Step 1.3.5.2

Reorder terms.

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

Step 1.3.6

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

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Multiply $\frac{1}{{(x + h)}^{2} + 1}$ by $1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 3

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

$\frac{1}{{(x + 0)}^{2} + 1}$

Step 4

Add $x$ and $0$ .

$\frac{1}{x^{2} + 1}$