Calculus Examples

$lim_{x \to π} \frac{x}{\cos (x)}$

Step 1

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

$\frac{lim_{x \to π} x}{lim_{x \to π} \cos (x)}$

Step 2

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

$\frac{lim_{x \to π} x}{\cos (lim_{x \to π} x)}$

Step 3

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

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Step 3.1

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

$\frac{π}{\cos (lim_{x \to π} x)}$

Step 3.2

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

$\frac{π}{\cos (π)}$

Step 4

Simplify the answer.

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Step 4.1

Separate fractions.

$\frac{π}{1} \cdot \frac{1}{\cos (π)}$

Step 4.2

Convert from $\frac{1}{\cos (π)}$ to $\sec (π)$ .

$\frac{π}{1} \sec (π)$

Step 4.3

Divide $π$ by $1$ .

$π \sec (π)$

Step 4.4

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

$π (- \sec (0))$

Step 4.5

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

$π (- 1 \cdot 1)$

Step 4.6

Multiply $- 1$ by $1$ .

$π \cdot - 1$

Step 4.7

Move $- 1$ to the left of $π$ .

$- 1 \cdot π$

Step 4.8

Rewrite $- 1 π$ as $- π$ .

$- π$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- π$

Decimal Form:

$- 3.14159265 \dots$