Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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Step 1.1.2.4.1

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

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

Step 1.1.2.4.2

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

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

Step 1.1.2.5

Simplify the answer.

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Step 1.1.2.5.1

Simplify each term.

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Step 1.1.2.5.1.1

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.

$\frac{π + π (- \sec (0))}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.5.1.2

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

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

Step 1.1.2.5.1.3

Multiply $- 1$ by $1$ .

$\frac{π + π \cdot - 1}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.5.1.4

Move $- 1$ to the left of $π$ .

$\frac{π - 1 \cdot π}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.5.1.5

Rewrite $- 1 π$ as $- π$ .

$\frac{π - π}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.5.2

Subtract $π$ from $π$ .

$\frac{0}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{0}{lim_{x \to π} x^{2} - lim_{x \to π} π^{2}}$

Step 1.1.3.1.2

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

$\frac{0}{{(lim_{x \to π} x)}^{2} - lim_{x \to π} π^{2}}$

Step 1.1.3.1.3

Evaluate the limit of $π^{2}$ which is constant as $x$ approaches $π$ .

$\frac{0}{{(lim_{x \to π} x)}^{2} - π^{2}}$

Step 1.1.3.2

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

$\frac{0}{π^{2} - π^{2}}$

Step 1.1.3.3

Subtract $π^{2}$ from $π^{2}$ .

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{x \to π} \frac{x + π \sec (x)}{x^{2} - π^{2}} = lim_{x \to π} \frac{\frac{d}{d x} [x + π \sec (x)]}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $x + π \sec (x)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [π \sec (x)]$ .

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

Step 1.3.3

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

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

Step 1.3.4

Evaluate $\frac{d}{d x} [π \sec (x)]$ .

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Step 1.3.4.1

Since $π$ is constant with respect to $x$ , the derivative of $π \sec (x)$ with respect to $x$ is $π \frac{d}{d x} [\sec (x)]$ .

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

Step 1.3.4.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$lim_{x \to π} \frac{1 + π \sec (x) \tan (x)}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.5

Reorder terms.

$lim_{x \to π} \frac{π \sec (x) \tan (x) + 1}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.6

By the Sum Rule, the derivative of $x^{2} - π^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- π^{2}]$ .

$lim_{x \to π} \frac{π \sec (x) \tan (x) + 1}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- π^{2}]}$

Step 1.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to π} \frac{π \sec (x) \tan (x) + 1}{2 x + \frac{d}{d x} [- π^{2}]}$

Step 1.3.8

Since $- π^{2}$ is constant with respect to $x$ , the derivative of $- π^{2}$ with respect to $x$ is $0$ .

$lim_{x \to π} \frac{π \sec (x) \tan (x) + 1}{2 x + 0}$

Step 1.3.9

Add $2 x$ and $0$ .

$lim_{x \to π} \frac{π \sec (x) \tan (x) + 1}{2 x}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to π} \frac{π \sec (x) \tan (x) + 1}{x}$

Step 2.2

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

$\frac{1}{2} \cdot \frac{lim_{x \to π} π \sec (x) \tan (x) + 1}{lim_{x \to π} x}$

Step 2.3

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

$\frac{1}{2} \cdot \frac{lim_{x \to π} π \sec (x) \tan (x) + lim_{x \to π} 1}{lim_{x \to π} x}$

Step 2.4

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

$\frac{1}{2} \cdot \frac{π lim_{x \to π} \sec (x) \tan (x) + lim_{x \to π} 1}{lim_{x \to π} x}$

Step 2.5

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

$\frac{1}{2} \cdot \frac{π lim_{x \to π} \sec (x) \cdot lim_{x \to π} \tan (x) + lim_{x \to π} 1}{lim_{x \to π} x}$

Step 2.6

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

$\frac{1}{2} \cdot \frac{π \sec (lim_{x \to π} x) \cdot lim_{x \to π} \tan (x) + lim_{x \to π} 1}{lim_{x \to π} x}$

Step 2.7

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

$\frac{1}{2} \cdot \frac{π \sec (lim_{x \to π} x) \cdot \tan (lim_{x \to π} x) + lim_{x \to π} 1}{lim_{x \to π} x}$

Step 2.8

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

$\frac{1}{2} \cdot \frac{π \sec (lim_{x \to π} x) \cdot \tan (lim_{x \to π} x) + 1}{lim_{x \to π} x}$

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 $x$ .

$\frac{1}{2} \cdot \frac{π \sec (π) \cdot \tan (lim_{x \to π} x) + 1}{lim_{x \to π} x}$

Step 3.2

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

$\frac{1}{2} \cdot \frac{π \sec (π) \cdot \tan (π) + 1}{lim_{x \to π} x}$

Step 3.3

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

$\frac{1}{2} \cdot \frac{π \sec (π) \cdot \tan (π) + 1}{π}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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.

$\frac{1}{2} \cdot \frac{π (- \sec (0)) \cdot \tan (π) + 1}{π}$

Step 4.1.2

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

$\frac{1}{2} \cdot \frac{π (- 1 \cdot 1) \cdot \tan (π) + 1}{π}$

Step 4.1.3

Multiply $- 1$ by $1$ .

$\frac{1}{2} \cdot \frac{π \cdot - 1 \cdot \tan (π) + 1}{π}$

Step 4.1.4

Move $- 1$ to the left of $π$ .

$\frac{1}{2} \cdot \frac{- 1 \cdot π \cdot \tan (π) + 1}{π}$

Step 4.1.5

Rewrite $- 1 π$ as $- π$ .

$\frac{1}{2} \cdot \frac{- π \cdot \tan (π) + 1}{π}$

Step 4.1.6

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

$\frac{1}{2} \cdot \frac{- π \cdot (- \tan (0)) + 1}{π}$

Step 4.1.7

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

$\frac{1}{2} \cdot \frac{- π \cdot (- 0) + 1}{π}$

Step 4.1.8

Multiply $- 1$ by $0$ .

$\frac{1}{2} \cdot \frac{- π \cdot 0 + 1}{π}$

Step 4.1.9

Multiply $- π \cdot 0$ .

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Step 4.1.9.1

Multiply $0$ by $- 1$ .

$\frac{1}{2} \cdot \frac{0 π + 1}{π}$

Step 4.1.9.2

Multiply $0$ by $π$ .

$\frac{1}{2} \cdot \frac{0 + 1}{π}$

Step 4.1.10

Add $0$ and $1$ .

$\frac{1}{2} \cdot \frac{1}{π}$

Step 4.2

Multiply $\frac{1}{2}$ by $\frac{1}{π}$ .

$\frac{1}{2 π}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2 π}$

Decimal Form:

$0.15915494 \dots$