Calculus Examples

$lim_{θ \to π} \frac{1 + \cos (θ)}{1 - \cos (θ)}$

Step 1

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

$\frac{lim_{θ \to π} 1 + \cos (θ)}{lim_{θ \to π} 1 - \cos (θ)}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $θ$ approaches $π$ .

$\frac{lim_{θ \to π} 1 + lim_{θ \to π} \cos (θ)}{lim_{θ \to π} 1 - \cos (θ)}$

Step 3

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

$\frac{1 + lim_{θ \to π} \cos (θ)}{lim_{θ \to π} 1 - \cos (θ)}$

Step 4

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

$\frac{1 + \cos (lim_{θ \to π} θ)}{lim_{θ \to π} 1 - \cos (θ)}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $θ$ approaches $π$ .

$\frac{1 + \cos (lim_{θ \to π} θ)}{lim_{θ \to π} 1 - lim_{θ \to π} \cos (θ)}$

Step 6

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

$\frac{1 + \cos (lim_{θ \to π} θ)}{1 - lim_{θ \to π} \cos (θ)}$

Step 7

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

$\frac{1 + \cos (lim_{θ \to π} θ)}{1 - \cos (lim_{θ \to π} θ)}$

Step 8

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

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Step 8.1

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

$\frac{1 + \cos (π)}{1 - \cos (lim_{θ \to π} θ)}$

Step 8.2

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

$\frac{1 + \cos (π)}{1 - \cos (π)}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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.

$\frac{1 - \cos (0)}{1 - \cos (π)}$

Step 9.1.2

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

$\frac{1 - 1 \cdot 1}{1 - \cos (π)}$

Step 9.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{1 - \cos (π)}$

Step 9.1.4

Subtract $1$ from $1$ .

$\frac{0}{1 - \cos (π)}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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.

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

Step 9.2.2

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

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

Step 9.2.3

Multiply $- (- 1 \cdot 1)$ .

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Step 9.2.3.1

Multiply $- 1$ by $1$ .

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

Step 9.2.3.2

Multiply $- 1$ by $- 1$ .

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

Step 9.2.4

Add $1$ and $1$ .

$\frac{0}{2}$

Step 9.3

Cancel the common factor of $0$ and $2$ .

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Step 9.3.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 0}{2 \cdot 1}$

Step 9.3.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 \cdot 1}$

Step 9.3.2.3

Rewrite the expression.

$\frac{0}{1}$

Step 9.3.2.4

Divide $0$ by $1$ .

$0$