Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

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

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

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

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

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

Step 1.1.3.2

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

$\frac{0}{\cos (0) - 1 \cdot 1}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

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

$\frac{0}{1 - 1 \cdot 1}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $1$ .

$\frac{0}{1 - 1}$

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 1.3.5

Subtract $\sec (x) \tan (x)$ from $0$ .

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 1.3.9

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

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

Step 1.4

Dividing two negative values results in a positive value.

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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Step 2.1.2.4.1

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

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

Step 2.1.2.4.2

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

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

Step 2.1.2.5

Simplify the answer.

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Step 2.1.2.5.1

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

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

Step 2.1.2.5.2

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

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

Step 2.1.2.5.3

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

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

$\frac{0}{0}$

Step 2.1.3.4

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sec (x)$ and $g (x) = \tan (x)$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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Step 2.3.4.1

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

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Step 2.3.4.1.1

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

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

Step 2.3.4.1.2

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

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

Step 2.3.4.2

Add $1$ and $2$ .

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

Step 2.3.5

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 2.3.6

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

$lim_{x \to 0} \frac{\sec^{3} (x) + \tan^{1} (x) \tan (x) \sec (x)}{\frac{d}{d x} [\sin (x)]}$

Step 2.3.7

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

$lim_{x \to 0} \frac{\sec^{3} (x) + \tan^{1} (x) \tan^{1} (x) \sec (x)}{\frac{d}{d x} [\sin (x)]}$

Step 2.3.8

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

$lim_{x \to 0} \frac{\sec^{3} (x) + {\tan (x)}^{1 + 1} \sec (x)}{\frac{d}{d x} [\sin (x)]}$

Step 2.3.9

Add $1$ and $1$ .

$lim_{x \to 0} \frac{\sec^{3} (x) + \tan^{2} (x) \sec (x)}{\frac{d}{d x} [\sin (x)]}$

Step 2.3.10

Reorder terms.

$lim_{x \to 0} \frac{\tan^{2} (x) \sec (x) + \sec^{3} (x)}{\frac{d}{d x} [\sin (x)]}$

Step 2.3.11

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

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Multiply $0$ by $1$ .

$\frac{0 + \sec^{3} (0)}{\cos (0)}$

Step 5.1.5

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

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

Step 5.1.6

One to any power is one.

$\frac{0 + 1}{\cos (0)}$

Step 5.1.7

Add $0$ and $1$ .

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

Step 5.2

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

$\frac{1}{1}$

Step 5.3

Cancel the common factor of $1$ .

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Step 5.3.1

Cancel the common factor.

$\frac{1}{1}$

Step 5.3.2

Rewrite the expression.

$1$