Calculus Examples

$lim_{h \to 0} \frac{7 h}{\sin (7 h)}$

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.

$\frac{lim_{h \to 0} 7 h}{lim_{h \to 0} \sin (7 h)}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Move the term $7$ outside of the limit because it is constant with respect to $h$ .

$\frac{7 lim_{h \to 0} h}{lim_{h \to 0} \sin (7 h)}$

Step 1.2.2

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

$\frac{7 \cdot 0}{lim_{h \to 0} \sin (7 h)}$

Step 1.2.3

Multiply $7$ by $0$ .

$\frac{0}{lim_{h \to 0} \sin (7 h)}$

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 limit inside the trig function because sine is continuous.

$\frac{0}{\sin (lim_{h \to 0} 7 h)}$

Step 1.3.1.2

Move the term $7$ outside of the limit because it is constant with respect to $h$ .

$\frac{0}{\sin (7 lim_{h \to 0} h)}$

Step 1.3.2

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

$\frac{0}{\sin (7 \cdot 0)}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Multiply $7$ by $0$ .

$\frac{0}{\sin (0)}$

Step 1.3.3.2

The exact value of $\sin (0)$ is $0$ .

$\frac{0}{0}$

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.

$lim_{h \to 0} \frac{7 h}{\sin (7 h)} = lim_{h \to 0} \frac{\frac{d}{d h} [7 h]}{\frac{d}{d h} [\sin (7 h)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [7 h]}{\frac{d}{d h} [\sin (7 h)]}$

Step 3.2

Since $7$ is constant with respect to $h$ , the derivative of $7 h$ with respect to $h$ is $7 \frac{d}{d h} [h]$ .

$lim_{h \to 0} \frac{7 \frac{d}{d h} [h]}{\frac{d}{d h} [\sin (7 h)]}$

Step 3.3

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

$lim_{h \to 0} \frac{7 \cdot 1}{\frac{d}{d h} [\sin (7 h)]}$

Step 3.4

Multiply $7$ by $1$ .

$lim_{h \to 0} \frac{7}{\frac{d}{d h} [\sin (7 h)]}$

Step 3.5

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = \sin (h)$ and $g (h) = 7 h$ .

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Step 3.5.1

To apply the Chain Rule, set $u$ as $7 h$ .

$lim_{h \to 0} \frac{7}{\frac{d}{d u} [\sin (u)] \frac{d}{d h} [7 h]}$

Step 3.5.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$lim_{h \to 0} \frac{7}{\cos (u) \frac{d}{d h} [7 h]}$

Step 3.5.3

Replace all occurrences of $u$ with $7 h$ .

$lim_{h \to 0} \frac{7}{\cos (7 h) \frac{d}{d h} [7 h]}$

Step 3.6

Since $7$ is constant with respect to $h$ , the derivative of $7 h$ with respect to $h$ is $7 \frac{d}{d h} [h]$ .

$lim_{h \to 0} \frac{7}{\cos (7 h) (7 \frac{d}{d h} [h])}$

Step 3.7

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

$lim_{h \to 0} \frac{7}{\cos (7 h) (7 \cdot 1)}$

Step 3.8

Multiply $7$ by $1$ .

$lim_{h \to 0} \frac{7}{\cos (7 h) \cdot 7}$

Step 3.9

Move $7$ to the left of $\cos (7 h)$ .

$lim_{h \to 0} \frac{7}{7 \cos (7 h)}$

Step 4

Cancel the common factor of $7$ .

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Step 4.1

Cancel the common factor.

$lim_{h \to 0} \frac{7}{7 \cos (7 h)}$

Step 4.2

Rewrite the expression.

$lim_{h \to 0} \frac{1}{\cos (7 h)}$

Step 5

Convert from $\frac{1}{\cos (7 h)}$ to $\sec (7 h)$ .

$lim_{h \to 0} \sec (7 h)$

Step 6

Evaluate the limit.

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Step 6.1

Move the limit inside the trig function because secant is continuous.

$\sec (lim_{h \to 0} 7 h)$

Step 6.2

Move the term $7$ outside of the limit because it is constant with respect to $h$ .

$\sec (7 lim_{h \to 0} h)$

Step 7

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

$\sec (7 \cdot 0)$

Step 8

Simplify the answer.

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Step 8.1

Multiply $7$ by $0$ .

$\sec (0)$

Step 8.2

The exact value of $\sec (0)$ is $1$ .

$1$