Calculus Examples

$lim_{x \to 0} \frac{5 x - \sin (5 x)}{5 x - \tan (5 x)}$

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.

$\frac{lim_{x \to 0} 5 x - \sin (5 x)}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

$\frac{lim_{x \to 0} 5 x - lim_{x \to 0} \sin (5 x)}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2.2

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to 0} x - lim_{x \to 0} \sin (5 x)}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2.3

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

$\frac{5 lim_{x \to 0} x - \sin (lim_{x \to 0} 5 x)}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2.4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to 0} x - \sin (5 lim_{x \to 0} x)}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2.5

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

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Step 1.2.5.1

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

$\frac{5 \cdot 0 - \sin (5 lim_{x \to 0} x)}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2.5.2

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

$\frac{5 \cdot 0 - \sin (5 \cdot 0)}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2.6

Simplify the answer.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

Multiply $5$ by $0$ .

$\frac{0 - \sin (5 \cdot 0)}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2.6.1.2

Multiply $5$ by $0$ .

$\frac{0 - \sin (0)}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2.6.1.3

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

$\frac{0 - 0}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2.6.1.4

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.2.6.2

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} 5 x - \tan (5 x)}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

$\frac{0}{lim_{x \to 0} 5 x - lim_{x \to 0} \tan (5 x)}$

Step 1.3.2

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{5 lim_{x \to 0} x - lim_{x \to 0} \tan (5 x)}$

Step 1.3.3

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

$\frac{0}{5 lim_{x \to 0} x - \tan (lim_{x \to 0} 5 x)}$

Step 1.3.4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{5 lim_{x \to 0} x - \tan (5 lim_{x \to 0} x)}$

Step 1.3.5

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

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Step 1.3.5.1

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

$\frac{0}{5 \cdot 0 - \tan (5 lim_{x \to 0} x)}$

Step 1.3.5.2

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

$\frac{0}{5 \cdot 0 - \tan (5 \cdot 0)}$

Step 1.3.6

Simplify the answer.

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Step 1.3.6.1

Simplify each term.

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Step 1.3.6.1.1

Multiply $5$ by $0$ .

$\frac{0}{0 - \tan (5 \cdot 0)}$

Step 1.3.6.1.2

Multiply $5$ by $0$ .

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

Step 1.3.6.1.3

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

$\frac{0}{0 - 0}$

Step 1.3.6.1.4

Multiply $- 1$ by $0$ .

$\frac{0}{0 + 0}$

Step 1.3.6.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.3.6.3

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

Undefined

$\frac{0}{0}$

Step 1.3.7

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.

$lim_{x \to 0} \frac{5 x - \sin (5 x)}{5 x - \tan (5 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [5 x - \sin (5 x)]}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [5 x - \sin (5 x)]}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.2

By the Sum Rule, the derivative of $5 x - \sin (5 x)$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- \sin (5 x)]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [5 x] + \frac{d}{d x} [- \sin (5 x)]}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.3

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 3.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{5 \frac{d}{d x} [x] + \frac{d}{d x} [- \sin (5 x)]}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.3.2

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

$lim_{x \to 0} \frac{5 \cdot 1 + \frac{d}{d x} [- \sin (5 x)]}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.3.3

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{5 + \frac{d}{d x} [- \sin (5 x)]}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.4

Evaluate $\frac{d}{d x} [- \sin (5 x)]$ .

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Step 3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (5 x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (5 x)]$ .

$lim_{x \to 0} \frac{5 - \frac{d}{d x} [\sin (5 x)]}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.4.2

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 $g (x) = 5 x$ .

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Step 3.4.2.1

To apply the Chain Rule, set $u$ as $5 x$ .

$lim_{x \to 0} \frac{5 - (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [5 x])}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.4.2.2

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

$lim_{x \to 0} \frac{5 - (\cos (u) \frac{d}{d x} [5 x])}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.4.2.3

Replace all occurrences of $u$ with $5 x$ .

$lim_{x \to 0} \frac{5 - (\cos (5 x) \frac{d}{d x} [5 x])}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.4.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{5 - (\cos (5 x) (5 \frac{d}{d x} [x]))}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.4.4

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

$lim_{x \to 0} \frac{5 - (\cos (5 x) (5 \cdot 1))}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.4.5

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{5 - (\cos (5 x) \cdot 5)}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.4.6

Move $5$ to the left of $\cos (5 x)$ .

$lim_{x \to 0} \frac{5 - (5 \cos (5 x))}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.4.7

Multiply $5$ by $- 1$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{\frac{d}{d x} [5 x - \tan (5 x)]}$

Step 3.5

By the Sum Rule, the derivative of $5 x - \tan (5 x)$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- \tan (5 x)]$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{\frac{d}{d x} [5 x] + \frac{d}{d x} [- \tan (5 x)]}$

Step 3.6

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 3.6.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 \frac{d}{d x} [x] + \frac{d}{d x} [- \tan (5 x)]}$

Step 3.6.2

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

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 \cdot 1 + \frac{d}{d x} [- \tan (5 x)]}$

Step 3.6.3

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 + \frac{d}{d x} [- \tan (5 x)]}$

Step 3.7

Evaluate $\frac{d}{d x} [- \tan (5 x)]$ .

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Step 3.7.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \tan (5 x)$ with respect to $x$ is $- \frac{d}{d x} [\tan (5 x)]$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 - \frac{d}{d x} [\tan (5 x)]}$

Step 3.7.2

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 $g (x) = 5 x$ .

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Step 3.7.2.1

To apply the Chain Rule, set $u$ as $5 x$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 - (\frac{d}{d u} [\tan (u)] \frac{d}{d x} [5 x])}$

Step 3.7.2.2

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

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 - (\sec^{2} (u) \frac{d}{d x} [5 x])}$

Step 3.7.2.3

Replace all occurrences of $u$ with $5 x$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 - (\sec^{2} (5 x) \frac{d}{d x} [5 x])}$

Step 3.7.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 - (\sec^{2} (5 x) (5 \frac{d}{d x} [x]))}$

Step 3.7.4

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

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 - (\sec^{2} (5 x) (5 \cdot 1))}$

Step 3.7.5

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 - (\sec^{2} (5 x) \cdot 5)}$

Step 3.7.6

Move $5$ to the left of $\sec^{2} (5 x)$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 - (5 \sec^{2} (5 x))}$

Step 3.7.7

Multiply $5$ by $- 1$ .

$lim_{x \to 0} \frac{5 - 5 \cos (5 x)}{5 - 5 \sec^{2} (5 x)}$

Step 4

Cancel the common factor of $5 - 5 \cos (5 x)$ and $5 - 5 \sec^{2} (5 x)$ .

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Step 4.1

Factor $5$ out of $5$ .

$lim_{x \to 0} \frac{5 (1) - 5 \cos (5 x)}{5 - 5 \sec^{2} (5 x)}$

Step 4.2

Factor $5$ out of $- 5 \cos (5 x)$ .

$lim_{x \to 0} \frac{5 (1) + 5 (- \cos (5 x))}{5 - 5 \sec^{2} (5 x)}$

Step 4.3

Factor $5$ out of $5 (1) + 5 (- \cos (5 x))$ .

$lim_{x \to 0} \frac{5 (1 - \cos (5 x))}{5 - 5 \sec^{2} (5 x)}$

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $5$ out of $5$ .

$lim_{x \to 0} \frac{5 (1 - \cos (5 x))}{5 \cdot 1 - 5 \sec^{2} (5 x)}$

Step 4.4.2

Factor $5$ out of $- 5 \sec^{2} (5 x)$ .

$lim_{x \to 0} \frac{5 (1 - \cos (5 x))}{5 \cdot 1 + 5 (- \sec^{2} (5 x))}$

Step 4.4.3

Factor $5$ out of $5 (1) + 5 (- \sec^{2} (5 x))$ .

$lim_{x \to 0} \frac{5 (1 - \cos (5 x))}{5 (1 - \sec^{2} (5 x))}$

Step 4.4.4

Cancel the common factor.

$lim_{x \to 0} \frac{5 (1 - \cos (5 x))}{5 (1 - \sec^{2} (5 x))}$

Step 4.4.5

Rewrite the expression.

$lim_{x \to 0} \frac{1 - \cos (5 x)}{1 - \sec^{2} (5 x)}$

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

$\frac{lim_{x \to 0} 1 - \cos (5 x)}{lim_{x \to 0} 1 - \sec^{2} (5 x)}$

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

Evaluate the limit.

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Step 5.1.2.1.1

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

$\frac{lim_{x \to 0} 1 - lim_{x \to 0} \cos (5 x)}{lim_{x \to 0} 1 - \sec^{2} (5 x)}$

Step 5.1.2.1.2

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

$\frac{1 - lim_{x \to 0} \cos (5 x)}{lim_{x \to 0} 1 - \sec^{2} (5 x)}$

Step 5.1.2.1.3

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

$\frac{1 - \cos (lim_{x \to 0} 5 x)}{lim_{x \to 0} 1 - \sec^{2} (5 x)}$

Step 5.1.2.1.4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 - \cos (5 lim_{x \to 0} x)}{lim_{x \to 0} 1 - \sec^{2} (5 x)}$

Step 5.1.2.2

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

$\frac{1 - \cos (5 \cdot 0)}{lim_{x \to 0} 1 - \sec^{2} (5 x)}$

Step 5.1.2.3

Simplify the answer.

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Step 5.1.2.3.1

Simplify each term.

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Step 5.1.2.3.1.1

Multiply $5$ by $0$ .

$\frac{1 - \cos (0)}{lim_{x \to 0} 1 - \sec^{2} (5 x)}$

Step 5.1.2.3.1.2

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

$\frac{1 - 1 \cdot 1}{lim_{x \to 0} 1 - \sec^{2} (5 x)}$

Step 5.1.2.3.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 0} 1 - \sec^{2} (5 x)}$

Step 5.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 0} 1 - \sec^{2} (5 x)}$

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

Evaluate the limit.

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Step 5.1.3.1.1

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

$\frac{0}{lim_{x \to 0} 1 - lim_{x \to 0} \sec^{2} (5 x)}$

Step 5.1.3.1.2

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

$\frac{0}{1 - lim_{x \to 0} \sec^{2} (5 x)}$

Step 5.1.3.1.3

Move the exponent $2$ from $\sec^{2} (5 x)$ outside the limit using the Limits Power Rule.

$\frac{0}{1 - {(lim_{x \to 0} \sec (5 x))}^{2}}$

Step 5.1.3.1.4

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

$\frac{0}{1 - \sec^{2} (lim_{x \to 0} 5 x)}$

Step 5.1.3.1.5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{1 - \sec^{2} (5 lim_{x \to 0} x)}$

Step 5.1.3.2

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

$\frac{0}{1 - \sec^{2} (5 \cdot 0)}$

Step 5.1.3.3

Simplify the answer.

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Step 5.1.3.3.1

Reorder $1$ and $- \sec^{2} (5 \cdot 0)$ .

$\frac{0}{- \sec^{2} (5 \cdot 0) + 1}$

Step 5.1.3.3.2

Factor $- 1$ out of $- \sec^{2} (5 \cdot 0)$ .

$\frac{0}{- (\sec^{2} (5 \cdot 0)) + 1}$

Step 5.1.3.3.3

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{0}{- \sec^{2} (5 \cdot 0) - 1 \cdot - 1}$

Step 5.1.3.3.4

Factor $- 1$ out of $- \sec^{2} (5 \cdot 0) - 1 \cdot - 1$ .

$\frac{0}{- (\sec^{2} (5 \cdot 0) - 1)}$

Step 5.1.3.3.5

Apply pythagorean identity.

$\frac{0}{- \tan^{2} (5 \cdot 0)}$

Step 5.1.3.3.6

Multiply $5$ by $0$ .

$\frac{0}{- \tan^{2} (0)}$

Step 5.1.3.3.7

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

$\frac{0}{- 0^{2}}$

Step 5.1.3.3.8

Raising $0$ to any positive power yields $0$ .

$\frac{0}{- 0}$

Step 5.1.3.3.9

Multiply $- 1$ by $0$ .

$\frac{0}{0}$

Step 5.1.3.3.10

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

Undefined

$\frac{0}{0}$

Step 5.1.3.4

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

Undefined

$\frac{0}{0}$

Step 5.1.4

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

Undefined

$\frac{0}{0}$

Step 5.2

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.

$lim_{x \to 0} \frac{1 - \cos (5 x)}{1 - \sec^{2} (5 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [1 - \cos (5 x)]}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [1 - \cos (5 x)]}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.2

By the Sum Rule, the derivative of $1 - \cos (5 x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (5 x)]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (5 x)]}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{0 + \frac{d}{d x} [- \cos (5 x)]}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.4

Evaluate $\frac{d}{d x} [- \cos (5 x)]$ .

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Step 5.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (5 x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (5 x)]$ .

$lim_{x \to 0} \frac{0 - \frac{d}{d x} [\cos (5 x)]}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 5 x$ .

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Step 5.3.4.2.1

To apply the Chain Rule, set $u$ as $5 x$ .

$lim_{x \to 0} \frac{0 - (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [5 x])}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.4.2.2

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

$lim_{x \to 0} \frac{0 - (- \sin (u) \frac{d}{d x} [5 x])}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.4.2.3

Replace all occurrences of $u$ with $5 x$ .

$lim_{x \to 0} \frac{0 - (- \sin (5 x) \frac{d}{d x} [5 x])}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.4.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{0 - (- \sin (5 x) (5 \frac{d}{d x} [x]))}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.4.4

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

$lim_{x \to 0} \frac{0 - (- \sin (5 x) (5 \cdot 1))}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.4.5

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{0 - (- \sin (5 x) \cdot 5)}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.4.6

Multiply $5$ by $- 1$ .

$lim_{x \to 0} \frac{0 - (- 5 \sin (5 x))}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.4.7

Multiply $- 5$ by $- 1$ .

$lim_{x \to 0} \frac{0 + 5 \sin (5 x)}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.5

Add $0$ and $5 \sin (5 x)$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{\frac{d}{d x} [1 - \sec^{2} (5 x)]}$

Step 5.3.6

By the Sum Rule, the derivative of $1 - \sec^{2} (5 x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \sec^{2} (5 x)]$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{\frac{d}{d x} [1] + \frac{d}{d x} [- \sec^{2} (5 x)]}$

Step 5.3.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 + \frac{d}{d x} [- \sec^{2} (5 x)]}$

Step 5.3.8

Evaluate $\frac{d}{d x} [- \sec^{2} (5 x)]$ .

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Step 5.3.8.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sec^{2} (5 x)$ with respect to $x$ is $- \frac{d}{d x} [\sec^{2} (5 x)]$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - \frac{d}{d x} [\sec^{2} (5 x)]}$

Step 5.3.8.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sec (5 x)$ .

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Step 5.3.8.2.1

To apply the Chain Rule, set $u_{1}$ as $\sec (5 x)$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sec (5 x)])}$

Step 5.3.8.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (2 u_{1} \frac{d}{d x} [\sec (5 x)])}$

Step 5.3.8.2.3

Replace all occurrences of $u_{1}$ with $\sec (5 x)$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (2 \sec (5 x) \frac{d}{d x} [\sec (5 x)])}$

Step 5.3.8.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 5 x$ .

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Step 5.3.8.3.1

To apply the Chain Rule, set $u_{2}$ as $5 x$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (2 \sec (5 x) (\frac{d}{d u_{2}} [\sec (u_{2})] \frac{d}{d x} [5 x]))}$

Step 5.3.8.3.2

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

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (2 \sec (5 x) (\sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [5 x]))}$

Step 5.3.8.3.3

Replace all occurrences of $u_{2}$ with $5 x$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (2 \sec (5 x) (\sec (5 x) \tan (5 x) \frac{d}{d x} [5 x]))}$

Step 5.3.8.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (2 \sec (5 x) (\sec (5 x) \tan (5 x) (5 \frac{d}{d x} [x])))}$

Step 5.3.8.5

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

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (2 \sec (5 x) (\sec (5 x) \tan (5 x) (5 \cdot 1)))}$

Step 5.3.8.6

Multiply $5$ by $1$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (2 \sec (5 x) (\sec (5 x) \tan (5 x) \cdot 5))}$

Step 5.3.8.7

Move $5$ to the left of $\sec (5 x) \tan (5 x)$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (2 \sec (5 x) (5 \cdot (\sec (5 x) \tan (5 x))))}$

Step 5.3.8.8

Multiply $5$ by $2$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (10 \sec (5 x) (\sec (5 x) \tan (5 x)))}$

Step 5.3.8.9

Raise $\sec (5 x)$ to the power of $1$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (10 (\sec^{1} (5 x) \sec (5 x)) \tan (5 x))}$

Step 5.3.8.10

Raise $\sec (5 x)$ to the power of $1$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (10 (\sec^{1} (5 x) \sec^{1} (5 x)) \tan (5 x))}$

Step 5.3.8.11

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

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (10 {\sec (5 x)}^{1 + 1} \tan (5 x))}$

Step 5.3.8.12

Add $1$ and $1$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - (10 \sec^{2} (5 x) \tan (5 x))}$

Step 5.3.8.13

Multiply $10$ by $- 1$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{0 - 10 \sec^{2} (5 x) \tan (5 x)}$

Step 5.3.9

Simplify.

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Step 5.3.9.1

Subtract $10 \sec^{2} (5 x) \tan (5 x)$ from $0$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{- 10 \sec^{2} (5 x) \tan (5 x)}$

Step 5.3.9.2

Rewrite $\sec (5 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{5 \sin (5 x)}{- 10 {(\frac{1}{\cos (5 x)})}^{2} \tan (5 x)}$

Step 5.3.9.3

Apply the product rule to $\frac{1}{\cos (5 x)}$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{- 10 \frac{1^{2}}{\cos^{2} (5 x)} \tan (5 x)}$

Step 5.3.9.4

One to any power is one.

$lim_{x \to 0} \frac{5 \sin (5 x)}{- 10 \frac{1}{\cos^{2} (5 x)} \tan (5 x)}$

Step 5.3.9.5

Combine $- 10$ and $\frac{1}{\cos^{2} (5 x)}$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{\frac{- 10}{\cos^{2} (5 x)} \tan (5 x)}$

Step 5.3.9.6

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{5 \sin (5 x)}{- \frac{10}{\cos^{2} (5 x)} \tan (5 x)}$

Step 5.3.9.7

Rewrite $\tan (5 x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{5 \sin (5 x)}{- \frac{10}{\cos^{2} (5 x)} \cdot \frac{\sin (5 x)}{\cos (5 x)}}$

Step 5.3.9.8

Multiply $- \frac{10}{\cos^{2} (5 x)} \cdot \frac{\sin (5 x)}{\cos (5 x)}$ .

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Step 5.3.9.8.1

Multiply $\frac{\sin (5 x)}{\cos (5 x)}$ by $\frac{10}{\cos^{2} (5 x)}$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{- \frac{\sin (5 x) \cdot 10}{\cos (5 x) \cos^{2} (5 x)}}$

Step 5.3.9.8.2

Multiply $\cos (5 x)$ by $\cos^{2} (5 x)$ by adding the exponents.

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Step 5.3.9.8.2.1

Multiply $\cos (5 x)$ by $\cos^{2} (5 x)$ .

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Step 5.3.9.8.2.1.1

Raise $\cos (5 x)$ to the power of $1$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{- \frac{\sin (5 x) \cdot 10}{\cos^{1} (5 x) \cos^{2} (5 x)}}$

Step 5.3.9.8.2.1.2

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

$lim_{x \to 0} \frac{5 \sin (5 x)}{- \frac{\sin (5 x) \cdot 10}{{\cos (5 x)}^{1 + 2}}}$

Step 5.3.9.8.2.2

Add $1$ and $2$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{- \frac{\sin (5 x) \cdot 10}{\cos^{3} (5 x)}}$

Step 5.3.9.9

Move $10$ to the left of $\sin (5 x)$ .

$lim_{x \to 0} \frac{5 \sin (5 x)}{- \frac{10 \sin (5 x)}{\cos^{3} (5 x)}}$

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} 5 \sin (5 x) (- \frac{\cos^{3} (5 x)}{10 \sin (5 x)})$

Step 5.5

Combine factors.

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Step 5.5.1

Multiply $- 1$ by $5$ .

$lim_{x \to 0} - 5 \sin (5 x) \frac{\cos^{3} (5 x)}{10 \sin (5 x)}$

Step 5.5.2

Combine $\frac{\cos^{3} (5 x)}{10 \sin (5 x)}$ and $- 5$ .

$lim_{x \to 0} \frac{\cos^{3} (5 x) \cdot - 5}{10 \sin (5 x)} \sin (5 x)$

Step 5.5.3

Combine $\frac{\cos^{3} (5 x) \cdot - 5}{10 \sin (5 x)}$ and $\sin (5 x)$ .

$lim_{x \to 0} \frac{\cos^{3} (5 x) \cdot - 5 \sin (5 x)}{10 \sin (5 x)}$

Step 5.6

Reduce.

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Step 5.6.1

Cancel the common factor of $- 5$ and $10$ .

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Step 5.6.1.1

Factor $5$ out of $\cos^{3} (5 x) \cdot - 5 \sin (5 x)$ .

$lim_{x \to 0} \frac{5 (\cos^{3} (5 x) \cdot - 1 \sin (5 x))}{10 \sin (5 x)}$

Step 5.6.1.2

Cancel the common factors.

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Step 5.6.1.2.1

Factor $5$ out of $10 \sin (5 x)$ .

$lim_{x \to 0} \frac{5 (\cos^{3} (5 x) \cdot - 1 \sin (5 x))}{5 (2 \sin (5 x))}$

Step 5.6.1.2.2

Cancel the common factor.

$lim_{x \to 0} \frac{5 (\cos^{3} (5 x) \cdot - 1 \sin (5 x))}{5 (2 \sin (5 x))}$

Step 5.6.1.2.3

Rewrite the expression.

$lim_{x \to 0} \frac{\cos^{3} (5 x) \cdot - 1 \sin (5 x)}{2 \sin (5 x)}$

Step 5.6.2

Cancel the common factor of $\sin (5 x)$ .

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Step 5.6.2.1

Cancel the common factor.

$lim_{x \to 0} \frac{\cos^{3} (5 x) \cdot - 1 \sin (5 x)}{2 \sin (5 x)}$

Step 5.6.2.2

Rewrite the expression.

$lim_{x \to 0} \frac{\cos^{3} (5 x) \cdot - 1}{2}$

Step 6

Evaluate the limit.

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Step 6.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 0} \cos^{3} (5 x) \cdot - 1$

Step 6.2

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot - 1 lim_{x \to 0} \cos^{3} (5 x)$

Step 6.3

Move the exponent $3$ from $\cos^{3} (5 x)$ outside the limit using the Limits Power Rule.

$\frac{1}{2} \cdot - 1 {(lim_{x \to 0} \cos (5 x))}^{3}$

Step 6.4

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

$\frac{1}{2} \cdot - 1 \cos^{3} (lim_{x \to 0} 5 x)$

Step 6.5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot - 1 \cos^{3} (5 lim_{x \to 0} x)$

Step 7

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

$\frac{1}{2} \cdot - 1 \cos^{3} (5 \cdot 0)$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{2}$ and $- 1$ .

$\frac{- 1}{2} \cos^{3} (5 \cdot 0)$

Step 8.2

Move the negative in front of the fraction.

$- \frac{1}{2} \cos^{3} (5 \cdot 0)$

Step 8.3

Multiply $5$ by $0$ .

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

Step 8.4

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

$- \frac{1}{2} \cdot 1^{3}$

Step 8.5

One to any power is one.

$- \frac{1}{2} \cdot 1$

Step 8.6

Multiply $- 1$ by $1$ .

$- \frac{1}{2}$