Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

$\frac{\sec (0)}{1 - \sin (0)}$

Step 7

Simplify the answer.

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Step 7.1

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

$\frac{1}{1 - \sin (0)}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

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

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

Step 7.2.2

Multiply $- 1$ by $0$ .

$\frac{1}{1 + 0}$

Step 7.2.3

Add $1$ and $0$ .

$\frac{1}{1}$

Step 7.3

Cancel the common factor of $1$ .

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Step 7.3.1

Cancel the common factor.

$\frac{1}{1}$

Step 7.3.2

Rewrite the expression.

$1$