Calculus Examples

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

Step 1

Apply trigonometric identities.

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Step 1.1

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

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

Step 1.2

Combine $\sin (x)$ and $\frac{1}{\cos (x)}$ .

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

Step 2

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

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

$\frac{0}{lim_{x \to 0} x}$

Step 3.1.3

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

$\frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{0}{0}$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

Divide $\sec^{2} (x)$ by $1$ .

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

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

$\sec^{2} (0)$

Step 6

Simplify the answer.

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Step 6.1

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

$1^{2}$

Step 6.2

One to any power is one.

$1$