Calculus Examples

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

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

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} 0}{lim_{h \to 0} h}

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

Step 1.1.2

Evaluate the limit of

0

$0$ which is constant as

h

$h$ approaches

0

$0$ .

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

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

Step 1.1.3

Evaluate the limit of

h

$h$ by plugging in

0

$0$ for

h

$h$ .

\frac{0}{0}

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by

0

$0$ . The expression is undefined.

Undefined

\frac{0}{0}

$\frac{0}{0}$

Step 1.2

Since

\frac{0}{0}

$\frac{0}{0}$ 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{0}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [0]}{\frac{d}{d h} [h]}

$lim_{h \to 0} \frac{0}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [0]}{\frac{d}{d h} [h]}$

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} [0]}{\frac{d}{d h} [h]}

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

Step 1.3.2

Since

0

$0$ is constant with respect to

h

$h$ , the derivative of

0

$0$ with respect to

h

$h$ is

0

$0$ .

lim_{h \to 0} \frac{0}{\frac{d}{d h} [h]}

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

Step 1.3.3

Differentiate using the Power Rule which states that

\frac{d}{d h} [h^{n}]

$\frac{d}{d h} [h^{n}]$ is

n h^{n - 1}

$n h^{n - 1}$ where

n = 1

$n = 1$ .

lim_{h \to 0} \frac{0}{1}

$lim_{h \to 0} \frac{0}{1}$

lim_{h \to 0} \frac{0}{1}

$lim_{h \to 0} \frac{0}{1}$

Step 1.4

Divide

0

$0$ by

1

$1$ .

lim_{h \to 0} 0

$lim_{h \to 0} 0$

lim_{h \to 0} 0

$lim_{h \to 0} 0$

Step 2

Evaluate the limit of

0

$0$ which is constant as

h

$h$ approaches

0

$0$ .

0

$0$