Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

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

$- \frac{1}{\sec^{2} (π)}$

Step 3

Simplify the answer.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$- \frac{1^{2}}{\sec^{2} (π)}$

Step 3.2

Rewrite $\frac{1^{2}}{\sec^{2} (π)}$ as ${(\frac{1}{\sec (π)})}^{2}$ .

$- {(\frac{1}{\sec (π)})}^{2}$

Step 3.3

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

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

Step 3.4

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

$- {(1 \cos (π))}^{2}$

Step 3.5

Multiply $\cos (π)$ by $1$ .

$- \cos^{2} (π)$

Step 3.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- {(- \cos (0))}^{2}$

Step 3.7

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

$- {(- 1 \cdot 1)}^{2}$

Step 3.8

Multiply $- 1$ by $1$ .

$- {(- 1)}^{2}$

Step 3.9

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.9.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 3.9.1.1

Raise $- 1$ to the power of $1$ .

${(- 1)}^{1} {(- 1)}^{2}$

Step 3.9.1.2

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

${(- 1)}^{1 + 2}$

Step 3.9.2

Add $1$ and $2$ .

${(- 1)}^{3}$

Step 3.10

Raise $- 1$ to the power of $3$ .

$- 1$