Calculus Examples

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

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (n x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{\sin (n x)}{\cos (n x)}}{\sin (x)}$

Step 1.2

Rewrite $\frac{\frac{\sin (n x)}{\cos (n x)}}{\sin (x)}$ as a product.

$lim_{x \to 0} \frac{\sin (n x)}{\cos (n x)} \cdot \frac{1}{\sin (x)}$

Step 1.3

Simplify.

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Step 1.3.1

Convert from $\frac{\sin (n x)}{\cos (n x)}$ to $\tan (n x)$ .

$lim_{x \to 0} \tan (n x) \frac{1}{\sin (x)}$

Step 1.3.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$lim_{x \to 0} \tan (n x) \csc (x)$

Step 2

Set up the limit as a left-sided limit.

$lim_{x \to 0^{-}} \tan (n x) \csc (x)$

Step 3

Evaluate the left-sided limit.

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Step 3.1

Rewrite $\tan (n x) \csc (x)$ as $\frac{\tan (n x)}{\frac{1}{\csc (x)}}$ .

$lim_{x \to 0^{-}} \frac{\tan (n x)}{\frac{1}{\csc (x)}}$

Step 3.2

Apply L'Hospital's rule.

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Step 3.2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 3.2.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0^{-}} \tan (n x)}{lim_{x \to 0^{-}} \frac{1}{\csc (x)}}$

Step 3.2.1.2

Evaluate the limit of the numerator.

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Step 3.2.1.2.1

Evaluate the limit.

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Step 3.2.1.2.1.1

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

$\frac{\tan (lim_{x \to 0^{-}} n x)}{lim_{x \to 0^{-}} \frac{1}{\csc (x)}}$

Step 3.2.1.2.1.2

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

$\frac{\tan (n lim_{x \to 0^{-}} x)}{lim_{x \to 0^{-}} \frac{1}{\csc (x)}}$

Step 3.2.1.2.2

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

$\frac{\tan (n \cdot 0)}{lim_{x \to 0^{-}} \frac{1}{\csc (x)}}$

Step 3.2.1.2.3

Simplify the answer.

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Step 3.2.1.2.3.1

Multiply $n$ by $0$ .

$\frac{\tan (0)}{lim_{x \to 0^{-}} \frac{1}{\csc (x)}}$

Step 3.2.1.2.3.2

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

$\frac{0}{lim_{x \to 0^{-}} \frac{1}{\csc (x)}}$

Step 3.2.1.3

Evaluate the limit of the denominator.

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Step 3.2.1.3.1

Apply trigonometric identities.

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Step 3.2.1.3.1.1

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{0}{lim_{x \to 0^{-}} \frac{1}{\frac{1}{\sin (x)}}}$

Step 3.2.1.3.1.2

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

$\frac{0}{lim_{x \to 0^{-}} 1 \sin (x)}$

Step 3.2.1.3.1.3

Multiply $\sin (x)$ by $1$ .

$\frac{0}{lim_{x \to 0^{-}} \sin (x)}$

Step 3.2.1.3.2

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

$\frac{0}{\sin (lim_{x \to 0^{-}} x)}$

Step 3.2.1.3.3

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

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

Step 3.2.1.3.4

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

$\frac{0}{0}$

Step 3.2.1.3.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.2.2

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.

$lim_{x \to 0^{-}} \frac{\tan (n x)}{\frac{1}{\csc (x)}} = lim_{x \to 0^{-}} \frac{\frac{d}{d x} [\tan (n x)]}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 3.2.3

Find the derivative of the numerator and denominator.

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Step 3.2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0^{-}} \frac{\frac{d}{d x} [\tan (n x)]}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 3.2.3.2

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

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Step 3.2.3.2.1

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

$lim_{x \to 0^{-}} \frac{\frac{d}{d u} [\tan (u)] \frac{d}{d x} [n x]}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 3.2.3.2.2

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

$lim_{x \to 0^{-}} \frac{\sec^{2} (u) \frac{d}{d x} [n x]}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 3.2.3.2.3

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

$lim_{x \to 0^{-}} \frac{\sec^{2} (n x) \frac{d}{d x} [n x]}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 3.2.3.3

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

$lim_{x \to 0^{-}} \frac{\sec^{2} (n x) (n \frac{d}{d x} [x])}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 3.2.3.4

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

$lim_{x \to 0^{-}} \frac{\sec^{2} (n x) (n \cdot 1)}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 3.2.3.5

Multiply $n$ by $1$ .

$lim_{x \to 0^{-}} \frac{\sec^{2} (n x) n}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 3.2.3.6

Reorder the factors of $\sec^{2} (n x) n$ .

$lim_{x \to 0^{-}} \frac{n \sec^{2} (n x)}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 3.2.3.7

Rewrite $\csc (x)$ in terms of sines and cosines.

$lim_{x \to 0^{-}} \frac{n \sec^{2} (n x)}{\frac{d}{d x} [\frac{1}{\frac{1}{\sin (x)}}]}$

Step 3.2.3.8

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

$lim_{x \to 0^{-}} \frac{n \sec^{2} (n x)}{\frac{d}{d x} [1 \sin (x)]}$

Step 3.2.3.9

Multiply $\sin (x)$ by $1$ .

$lim_{x \to 0^{-}} \frac{n \sec^{2} (n x)}{\frac{d}{d x} [\sin (x)]}$

Step 3.2.3.10

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

$lim_{x \to 0^{-}} \frac{n \sec^{2} (n x)}{\cos (x)}$

Step 3.2.4

Simplify the numerator.

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Step 3.2.4.1

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

$lim_{x \to 0^{-}} \frac{n {(\frac{1}{\cos (n x)})}^{2}}{\cos (x)}$

Step 3.2.4.2

Apply the product rule to $\frac{1}{\cos (n x)}$ .

$lim_{x \to 0^{-}} \frac{n \frac{1^{2}}{\cos^{2} (n x)}}{\cos (x)}$

Step 3.2.4.3

One to any power is one.

$lim_{x \to 0^{-}} \frac{n \frac{1}{\cos^{2} (n x)}}{\cos (x)}$

Step 3.2.5

Combine $n$ and $\frac{1}{\cos^{2} (n x)}$ .

$lim_{x \to 0^{-}} \frac{\frac{n}{\cos^{2} (n x)}}{\cos (x)}$

Step 3.2.6

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0^{-}} \frac{n}{\cos^{2} (n x)} \cdot \frac{1}{\cos (x)}$

Step 3.2.7

Combine.

$lim_{x \to 0^{-}} \frac{n \cdot 1}{\cos^{2} (n x) \cos (x)}$

Step 3.2.8

Multiply $n$ by $1$ .

$lim_{x \to 0^{-}} \frac{n}{\cos^{2} (n x) \cos (x)}$

Step 3.2.9

Multiply by $1$ .

$lim_{x \to 0^{-}} \frac{n}{\cos^{2} (n x) \cdot 1 \cos (x)}$

Step 3.2.10

Separate fractions.

$lim_{x \to 0^{-}} \frac{n}{1 \cos (x)} \cdot \frac{1}{\cos^{2} (n x)}$

Step 3.2.11

Convert from $\frac{1}{\cos^{2} (n x)}$ to $\sec^{2} (n x)$ .

$lim_{x \to 0^{-}} \frac{n}{1 \cos (x)} \sec^{2} (n x)$

Step 3.2.12

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

$lim_{x \to 0^{-}} \frac{n}{\cos (x)} \sec^{2} (n x)$

Step 3.2.13

Separate fractions.

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

Step 3.2.14

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

$lim_{x \to 0^{-}} \frac{n}{1} \sec (x) \sec^{2} (n x)$

Step 3.2.15

Divide $n$ by $1$ .

$lim_{x \to 0^{-}} n \sec (x) \sec^{2} (n x)$

Step 3.3

Evaluate the limit.

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Step 3.3.1

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

$n lim_{x \to 0^{-}} \sec (x) \sec^{2} (n x)$

Step 3.3.2

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

$n lim_{x \to 0^{-}} \sec (x) \cdot lim_{x \to 0^{-}} \sec^{2} (n x)$

Step 3.3.3

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

$n \sec (lim_{x \to 0^{-}} x) \cdot lim_{x \to 0^{-}} \sec^{2} (n x)$

Step 3.3.4

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

$n \sec (lim_{x \to 0^{-}} x) \cdot {(lim_{x \to 0^{-}} \sec (n x))}^{2}$

Step 3.3.5

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

$n \sec (lim_{x \to 0^{-}} x) \cdot \sec^{2} (lim_{x \to 0^{-}} n x)$

Step 3.3.6

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

$n \sec (lim_{x \to 0^{-}} x) \cdot \sec^{2} (n lim_{x \to 0^{-}} x)$

Step 3.4

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

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Step 3.4.1

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

$n \sec (0) \cdot \sec^{2} (n lim_{x \to 0^{-}} x)$

Step 3.4.2

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

$n \sec (0) \cdot \sec^{2} (n \cdot 0)$

Step 3.5

Simplify the answer.

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Step 3.5.1

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

$n \cdot 1 \cdot \sec^{2} (n \cdot 0)$

Step 3.5.2

Multiply $n$ by $1$ .

$n \cdot \sec^{2} (n \cdot 0)$

Step 3.5.3

Multiply $n$ by $0$ .

$n \cdot \sec^{2} (0)$

Step 3.5.4

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

$n \cdot 1^{2}$

Step 3.5.5

One to any power is one.

$n \cdot 1$

Step 3.5.6

Multiply $n$ by $1$ .

$n$

Step 4

Set up the limit as a right-sided limit.

$lim_{x \to 0^{+}} \tan (n x) \csc (x)$

Step 5

Evaluate the right-sided limit.

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Step 5.1

Rewrite $\tan (n x) \csc (x)$ as $\frac{\tan (n x)}{\frac{1}{\csc (x)}}$ .

$lim_{x \to 0^{+}} \frac{\tan (n x)}{\frac{1}{\csc (x)}}$

Step 5.2

Apply L'Hospital's rule.

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Step 5.2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 5.2.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0^{+}} \tan (n x)}{lim_{x \to 0^{+}} \frac{1}{\csc (x)}}$

Step 5.2.1.2

Evaluate the limit of the numerator.

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Step 5.2.1.2.1

Evaluate the limit.

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Step 5.2.1.2.1.1

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

$\frac{\tan (lim_{x \to 0^{+}} n x)}{lim_{x \to 0^{+}} \frac{1}{\csc (x)}}$

Step 5.2.1.2.1.2

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

$\frac{\tan (n lim_{x \to 0^{+}} x)}{lim_{x \to 0^{+}} \frac{1}{\csc (x)}}$

Step 5.2.1.2.2

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

$\frac{\tan (n \cdot 0)}{lim_{x \to 0^{+}} \frac{1}{\csc (x)}}$

Step 5.2.1.2.3

Simplify the answer.

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Step 5.2.1.2.3.1

Multiply $n$ by $0$ .

$\frac{\tan (0)}{lim_{x \to 0^{+}} \frac{1}{\csc (x)}}$

Step 5.2.1.2.3.2

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

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

Step 5.2.1.3

Evaluate the limit of the denominator.

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Step 5.2.1.3.1

Apply trigonometric identities.

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Step 5.2.1.3.1.1

Rewrite $\csc (x)$ in terms of sines and cosines.

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

Step 5.2.1.3.1.2

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

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

Step 5.2.1.3.1.3

Multiply $\sin (x)$ by $1$ .

$\frac{0}{lim_{x \to 0^{+}} \sin (x)}$

Step 5.2.1.3.2

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

$\frac{0}{\sin (lim_{x \to 0^{+}} x)}$

Step 5.2.1.3.3

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

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

Step 5.2.1.3.4

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

$\frac{0}{0}$

Step 5.2.1.3.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 5.2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 5.2.2

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.

$lim_{x \to 0^{+}} \frac{\tan (n x)}{\frac{1}{\csc (x)}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\tan (n x)]}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 5.2.3

Find the derivative of the numerator and denominator.

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Step 5.2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\tan (n x)]}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 5.2.3.2

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

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Step 5.2.3.2.1

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

$lim_{x \to 0^{+}} \frac{\frac{d}{d u} [\tan (u)] \frac{d}{d x} [n x]}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 5.2.3.2.2

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

$lim_{x \to 0^{+}} \frac{\sec^{2} (u) \frac{d}{d x} [n x]}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 5.2.3.2.3

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

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

Step 5.2.3.3

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

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

Step 5.2.3.4

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

$lim_{x \to 0^{+}} \frac{\sec^{2} (n x) (n \cdot 1)}{\frac{d}{d x} [\frac{1}{\csc (x)}]}$

Step 5.2.3.5

Multiply $n$ by $1$ .

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

Step 5.2.3.6

Reorder the factors of $\sec^{2} (n x) n$ .

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

Step 5.2.3.7

Rewrite $\csc (x)$ in terms of sines and cosines.

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

Step 5.2.3.8

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

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

Step 5.2.3.9

Multiply $\sin (x)$ by $1$ .

$lim_{x \to 0^{+}} \frac{n \sec^{2} (n x)}{\frac{d}{d x} [\sin (x)]}$

Step 5.2.3.10

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

$lim_{x \to 0^{+}} \frac{n \sec^{2} (n x)}{\cos (x)}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

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

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

Step 5.2.4.2

Apply the product rule to $\frac{1}{\cos (n x)}$ .

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

Step 5.2.4.3

One to any power is one.

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

Step 5.2.5

Combine $n$ and $\frac{1}{\cos^{2} (n x)}$ .

$lim_{x \to 0^{+}} \frac{\frac{n}{\cos^{2} (n x)}}{\cos (x)}$

Step 5.2.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.7

Combine.

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

Step 5.2.8

Multiply $n$ by $1$ .

$lim_{x \to 0^{+}} \frac{n}{\cos^{2} (n x) \cos (x)}$

Step 5.2.9

Multiply by $1$ .

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

Step 5.2.10

Separate fractions.

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

Step 5.2.11

Convert from $\frac{1}{\cos^{2} (n x)}$ to $\sec^{2} (n x)$ .

$lim_{x \to 0^{+}} \frac{n}{1 \cos (x)} \sec^{2} (n x)$

Step 5.2.12

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

$lim_{x \to 0^{+}} \frac{n}{\cos (x)} \sec^{2} (n x)$

Step 5.2.13

Separate fractions.

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

Step 5.2.14

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

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

Step 5.2.15

Divide $n$ by $1$ .

$lim_{x \to 0^{+}} n \sec (x) \sec^{2} (n x)$

Step 5.3

Evaluate the limit.

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Step 5.3.1

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

$n lim_{x \to 0^{+}} \sec (x) \sec^{2} (n x)$

Step 5.3.2

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

$n lim_{x \to 0^{+}} \sec (x) \cdot lim_{x \to 0^{+}} \sec^{2} (n x)$

Step 5.3.3

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

$n \sec (lim_{x \to 0^{+}} x) \cdot lim_{x \to 0^{+}} \sec^{2} (n x)$

Step 5.3.4

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

$n \sec (lim_{x \to 0^{+}} x) \cdot {(lim_{x \to 0^{+}} \sec (n x))}^{2}$

Step 5.3.5

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

$n \sec (lim_{x \to 0^{+}} x) \cdot \sec^{2} (lim_{x \to 0^{+}} n x)$

Step 5.3.6

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

$n \sec (lim_{x \to 0^{+}} x) \cdot \sec^{2} (n lim_{x \to 0^{+}} x)$

Step 5.4

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

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Step 5.4.1

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

$n \sec (0) \cdot \sec^{2} (n lim_{x \to 0^{+}} x)$

Step 5.4.2

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

$n \sec (0) \cdot \sec^{2} (n \cdot 0)$

Step 5.5

Simplify the answer.

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Step 5.5.1

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

$n \cdot 1 \cdot \sec^{2} (n \cdot 0)$

Step 5.5.2

Multiply $n$ by $1$ .

$n \cdot \sec^{2} (n \cdot 0)$

Step 5.5.3

Multiply $n$ by $0$ .

$n \cdot \sec^{2} (0)$

Step 5.5.4

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

$n \cdot 1^{2}$

Step 5.5.5

One to any power is one.

$n \cdot 1$

Step 5.5.6

Multiply $n$ by $1$ .

$n$

Step 6

Since the left-sided limit is equal to the right-sided limit, the limit is equal to $n$ .

$n$