Calculus Examples

$lim_{x \to 0} \frac{\sin (7 x)}{\tan (3 x)}$

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.

$\frac{lim_{x \to 0} \sin (7 x)}{lim_{x \to 0} \tan (3 x)}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

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

$\frac{\sin (lim_{x \to 0} 7 x)}{lim_{x \to 0} \tan (3 x)}$

Step 1.2.1.2

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

$\frac{\sin (7 lim_{x \to 0} x)}{lim_{x \to 0} \tan (3 x)}$

Step 1.2.2

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

$\frac{\sin (7 \cdot 0)}{lim_{x \to 0} \tan (3 x)}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Multiply $7$ by $0$ .

$\frac{\sin (0)}{lim_{x \to 0} \tan (3 x)}$

Step 1.2.3.2

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

$\frac{0}{lim_{x \to 0} \tan (3 x)}$

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 tangent is continuous.

$\frac{0}{\tan (lim_{x \to 0} 3 x)}$

Step 1.3.1.2

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

$\frac{0}{\tan (3 lim_{x \to 0} x)}$

Step 1.3.2

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

$\frac{0}{\tan (3 \cdot 0)}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Multiply $3$ by $0$ .

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

Step 1.3.3.2

The exact value of $\tan (0)$ 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.

$lim_{x \to 0} \frac{\sin (7 x)}{\tan (3 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (7 x)]}{\frac{d}{d x} [\tan (3 x)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sin (7 x)]}{\frac{d}{d x} [\tan (3 x)]}$

Step 3.2

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

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Step 3.2.1

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

$lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [7 x]}{\frac{d}{d x} [\tan (3 x)]}$

Step 3.2.2

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

$lim_{x \to 0} \frac{\cos (u) \frac{d}{d x} [7 x]}{\frac{d}{d x} [\tan (3 x)]}$

Step 3.2.3

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

$lim_{x \to 0} \frac{\cos (7 x) \frac{d}{d x} [7 x]}{\frac{d}{d x} [\tan (3 x)]}$

Step 3.3

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

$lim_{x \to 0} \frac{\cos (7 x) (7 \frac{d}{d x} [x])}{\frac{d}{d x} [\tan (3 x)]}$

Step 3.4

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

$lim_{x \to 0} \frac{\cos (7 x) (7 \cdot 1)}{\frac{d}{d x} [\tan (3 x)]}$

Step 3.5

Multiply $7$ by $1$ .

$lim_{x \to 0} \frac{\cos (7 x) \cdot 7}{\frac{d}{d x} [\tan (3 x)]}$

Step 3.6

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

$lim_{x \to 0} \frac{7 \cdot \cos (7 x)}{\frac{d}{d x} [\tan (3 x)]}$

Step 3.7

Multiply $7$ by $\cos (7 x)$ .

$lim_{x \to 0} \frac{7 \cos (7 x)}{\frac{d}{d x} [\tan (3 x)]}$

Step 3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 3 x$ .

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Step 3.8.1

To apply the Chain Rule, set $u$ as $3 x$ .

$lim_{x \to 0} \frac{7 \cos (7 x)}{\frac{d}{d u} [\tan (u)] \frac{d}{d x} [3 x]}$

Step 3.8.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$lim_{x \to 0} \frac{7 \cos (7 x)}{\sec^{2} (u) \frac{d}{d x} [3 x]}$

Step 3.8.3

Replace all occurrences of $u$ with $3 x$ .

$lim_{x \to 0} \frac{7 \cos (7 x)}{\sec^{2} (3 x) \frac{d}{d x} [3 x]}$

Step 3.9

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

$lim_{x \to 0} \frac{7 \cos (7 x)}{\sec^{2} (3 x) (3 \frac{d}{d x} [x])}$

Step 3.10

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

$lim_{x \to 0} \frac{7 \cos (7 x)}{\sec^{2} (3 x) (3 \cdot 1)}$

Step 3.11

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{7 \cos (7 x)}{\sec^{2} (3 x) \cdot 3}$

Step 3.12

Move $3$ to the left of $\sec^{2} (3 x)$ .

$lim_{x \to 0} \frac{7 \cos (7 x)}{3 \cdot \sec^{2} (3 x)}$

Step 3.13

Multiply $3$ by $\sec^{2} (3 x)$ .

$lim_{x \to 0} \frac{7 \cos (7 x)}{3 \sec^{2} (3 x)}$

Step 4

Move the term $\frac{7}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{7}{3} lim_{x \to 0} \frac{\cos (7 x)}{\sec^{2} (3 x)}$

Step 5

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

$\frac{7}{3} \cdot \frac{lim_{x \to 0} \cos (7 x)}{lim_{x \to 0} \sec^{2} (3 x)}$

Step 6

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

$\frac{7}{3} \cdot \frac{\cos (lim_{x \to 0} 7 x)}{lim_{x \to 0} \sec^{2} (3 x)}$

Step 7

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

$\frac{7}{3} \cdot \frac{\cos (7 lim_{x \to 0} x)}{lim_{x \to 0} \sec^{2} (3 x)}$

Step 8

Move the exponent $2$ from $\sec^{2} (3 x)$ outside the limit using the Limits Power Rule.

$\frac{7}{3} \cdot \frac{\cos (7 lim_{x \to 0} x)}{{(lim_{x \to 0} \sec (3 x))}^{2}}$

Step 9

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

$\frac{7}{3} \cdot \frac{\cos (7 lim_{x \to 0} x)}{\sec^{2} (lim_{x \to 0} 3 x)}$

Step 10

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

$\frac{7}{3} \cdot \frac{\cos (7 lim_{x \to 0} x)}{\sec^{2} (3 lim_{x \to 0} x)}$

Step 11

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

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Step 11.1

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

$\frac{7}{3} \cdot \frac{\cos (7 \cdot 0)}{\sec^{2} (3 lim_{x \to 0} x)}$

Step 11.2

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

$\frac{7}{3} \cdot \frac{\cos (7 \cdot 0)}{\sec^{2} (3 \cdot 0)}$

Step 12

Simplify the answer.

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Step 12.1

Combine.

$\frac{7 \cos (7 \cdot 0)}{3 \sec^{2} (3 \cdot 0)}$

Step 12.2

Factor $\sec (3 \cdot 0)$ out of $\sec^{2} (3 \cdot 0)$ .

$\frac{7 \cos (7 \cdot 0)}{3 (\sec (3 \cdot 0) \sec (3 \cdot 0))}$

Step 12.3

Separate fractions.

$\frac{7}{3 (\sec (3 \cdot 0))} \cdot \frac{\cos (7 \cdot 0)}{\sec (3 \cdot 0)}$

Step 12.4

Rewrite $\sec (3 \cdot 0)$ in terms of sines and cosines.

$\frac{7}{3 (\sec (3 \cdot 0))} \cdot \frac{\cos (7 \cdot 0)}{\frac{1}{\cos (3 \cdot 0)}}$

Step 12.5

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (3 \cdot 0)}$ .

$\frac{7}{3 (\sec (3 \cdot 0))} (\cos (7 \cdot 0) \cos (3 \cdot 0))$

Step 12.6

Multiply $3$ by $0$ .

$\frac{7}{3 \sec (0)} (\cos (7 \cdot 0) \cos (3 \cdot 0))$

Step 12.7

Multiply $7$ by $0$ .

$\frac{7}{3 \sec (0)} (\cos (0) \cos (3 \cdot 0))$

Step 12.8

Multiply $3$ by $0$ .

$\frac{7}{3 \sec (0)} (\cos (0) \cos (0))$

Step 12.9

Separate fractions.

$\frac{7}{3} \cdot \frac{1}{\sec (0)} (\cos (0) \cos (0))$

Step 12.10

Rewrite $\sec (0)$ in terms of sines and cosines.

$\frac{7}{3} \cdot \frac{1}{\frac{1}{\cos (0)}} (\cos (0) \cos (0))$

Step 12.11

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (0)}$ .

$\frac{7}{3} (1 \cos (0)) (\cos (0) \cos (0))$

Step 12.12

Multiply $\cos (0)$ by $1$ .

$\frac{7}{3} \cos (0) (\cos (0) \cos (0))$

Step 12.13

Multiply $\cos (0)$ by $\cos (0)$ by adding the exponents.

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Step 12.13.1

Move $\cos (0)$ .

$\frac{7}{3} (\cos (0) \cos (0)) \cos (0)$

Step 12.13.2

Multiply $\cos (0)$ by $\cos (0)$ .

$\frac{7}{3} \cos^{2} (0) \cos (0)$

Step 12.14

Multiply $\cos^{2} (0)$ by $\cos (0)$ by adding the exponents.

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Step 12.14.1

Move $\cos (0)$ .

$\frac{7}{3} (\cos (0) \cos^{2} (0))$

Step 12.14.2

Multiply $\cos (0)$ by $\cos^{2} (0)$ .

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Step 12.14.2.1

Raise $\cos (0)$ to the power of $1$ .

$\frac{7}{3} (\cos^{1} (0) \cos^{2} (0))$

Step 12.14.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{7}{3} {\cos (0)}^{1 + 2}$

Step 12.14.3

Add $1$ and $2$ .

$\frac{7}{3} \cos^{3} (0)$

Step 12.15

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

$\frac{7}{3} \cdot 1^{3}$

Step 12.16

One to any power is one.

$\frac{7}{3} \cdot 1$

Step 12.17

Multiply $\frac{7}{3}$ by $1$ .

$\frac{7}{3}$