Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

One to any power is one.

$\frac{1 - 1 \cdot 1}{lim_{x \to 1} \sin (π x)}$

Step 1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 1} \sin (π x)}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

$\frac{0}{\sin (lim_{x \to 1} π x)}$

Step 1.3.1.2

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

$\frac{0}{\sin (π lim_{x \to 1} x)}$

Step 1.3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{\sin (π \cdot 1)}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Multiply $π$ by $1$ .

$\frac{0}{\sin (π)}$

Step 1.3.3.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 1.3.3.3

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

$\frac{0}{0}$

Step 1.3.3.4

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 3.3

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

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

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{2 x + 0}{\frac{d}{d x} [\sin (π x)]}$

Step 3.5

Add $2 x$ and $0$ .

$lim_{x \to 1} \frac{2 x}{\frac{d}{d x} [\sin (π x)]}$

Step 3.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = π x$ .

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Step 3.6.1

To apply the Chain Rule, set $u$ as $π x$ .

$lim_{x \to 1} \frac{2 x}{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [π x]}$

Step 3.6.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$lim_{x \to 1} \frac{2 x}{\cos (u) \frac{d}{d x} [π x]}$

Step 3.6.3

Replace all occurrences of $u$ with $π x$ .

$lim_{x \to 1} \frac{2 x}{\cos (π x) \frac{d}{d x} [π x]}$

Step 3.7

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

$lim_{x \to 1} \frac{2 x}{\cos (π x) (π \frac{d}{d x} [x])}$

Step 3.8

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

$lim_{x \to 1} \frac{2 x}{\cos (π x) (π \cdot 1)}$

Step 3.9

Multiply $π$ by $1$ .

$lim_{x \to 1} \frac{2 x}{\cos (π x) π}$

Step 3.10

Reorder the factors of $\cos (π x) π$ .

$lim_{x \to 1} \frac{2 x}{π \cos (π x)}$

Step 4

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

$\frac{2}{π} lim_{x \to 1} \frac{x}{\cos (π x)}$

Step 5

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{2}{π} \cdot \frac{lim_{x \to 1} x}{lim_{x \to 1} \cos (π x)}$

Step 6

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

$\frac{2}{π} \cdot \frac{lim_{x \to 1} x}{\cos (lim_{x \to 1} π x)}$

Step 7

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

$\frac{2}{π} \cdot \frac{lim_{x \to 1} x}{\cos (π lim_{x \to 1} x)}$

Step 8

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{2}{π} \cdot \frac{1}{\cos (π lim_{x \to 1} x)}$

Step 8.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{2}{π} \cdot \frac{1}{\cos (π \cdot 1)}$

Step 9

Simplify the answer.

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Step 9.1

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

$\frac{2}{π} \sec (π \cdot 1)$

Step 9.2

Multiply $π$ by $1$ .

$\frac{2}{π} \sec (π)$

Step 9.3

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.

$\frac{2}{π} (- \sec (0))$

Step 9.4

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

$\frac{2}{π} (- 1 \cdot 1)$

Step 9.5

Multiply $- 1$ by $1$ .

$\frac{2}{π} \cdot - 1$

Step 9.6

Multiply $\frac{2}{π} \cdot - 1$ .

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Step 9.6.1

Combine $\frac{2}{π}$ and $- 1$ .

$\frac{2 \cdot - 1}{π}$

Step 9.6.2

Multiply $2$ by $- 1$ .

$\frac{- 2}{π}$

Step 9.7

Move the negative in front of the fraction.

$- \frac{2}{π}$