Calculus Examples

$lim_{x \to π} \frac{2 x^{2} - 6 x π + 4 π^{2}}{x^{2} - π^{2}}$

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.

$\frac{lim_{x \to π} 2 x^{2} - 6 x π + 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

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 $π$ .

$\frac{lim_{x \to π} 2 x^{2} - lim_{x \to π} 6 x π + lim_{x \to π} 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to π} x^{2} - lim_{x \to π} 6 x π + lim_{x \to π} 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.3

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

$\frac{2 {(lim_{x \to π} x)}^{2} - lim_{x \to π} 6 x π + lim_{x \to π} 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.4

Move the term $6 π$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 {(lim_{x \to π} x)}^{2} - 1 \cdot 6 π lim_{x \to π} x + lim_{x \to π} 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.5

Evaluate the limit of $4 π^{2}$ which is constant as $x$ approaches $π$ .

$\frac{2 {(lim_{x \to π} x)}^{2} - 1 \cdot 6 π lim_{x \to π} x + 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

$\frac{2 π^{2} - 1 \cdot 6 π lim_{x \to π} x + 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.6.2

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

$\frac{2 π^{2} - 1 \cdot 6 π π + 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Simplify each term.

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Step 1.1.2.7.1.1

Multiply $- 1$ by $6$ .

$\frac{2 π^{2} - 6 π π + 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.7.1.2

Multiply $- 6 π π$ .

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Step 1.1.2.7.1.2.1

Raise $π$ to the power of $1$ .

$\frac{2 π^{2} - 6 (π^{1} π) + 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.7.1.2.2

Raise $π$ to the power of $1$ .

$\frac{2 π^{2} - 6 (π^{1} π^{1}) + 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.7.1.2.3

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

$\frac{2 π^{2} - 6 π^{1 + 1} + 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.7.1.2.4

Add $1$ and $1$ .

$\frac{2 π^{2} - 6 π^{2} + 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.7.2

Subtract $6 π^{2}$ from $2 π^{2}$ .

$\frac{- 4 π^{2} + 4 π^{2}}{lim_{x \to π} x^{2} - π^{2}}$

Step 1.1.2.7.3

Add $- 4 π^{2}$ and $4 π^{2}$ .

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

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.

$lim_{x \to π} \frac{2 x^{2} - 6 x π + 4 π^{2}}{x^{2} - π^{2}} = lim_{x \to π} \frac{\frac{d}{d x} [2 x^{2} - 6 x π + 4 π^{2}]}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to π} \frac{\frac{d}{d x} [2 x^{2} - 6 x π + 4 π^{2}]}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.2

By the Sum Rule, the derivative of $2 x^{2} - 6 x π + 4 π^{2}$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 6 x π] + \frac{d}{d x} [4 π^{2}]$ .

$lim_{x \to π} \frac{\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 6 x π] + \frac{d}{d x} [4 π^{2}]}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [2 x^{2}]$ .

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Step 1.3.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{2}$ with respect to $x$ is $2 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to π} \frac{2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x π] + \frac{d}{d x} [4 π^{2}]}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.3.2

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

$lim_{x \to π} \frac{2 (2 x) + \frac{d}{d x} [- 6 x π] + \frac{d}{d x} [4 π^{2}]}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.3.3

Multiply $2$ by $2$ .

$lim_{x \to π} \frac{4 x + \frac{d}{d x} [- 6 x π] + \frac{d}{d x} [4 π^{2}]}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- 6 x π]$ .

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Step 1.3.4.1

Since $- 6 π$ is constant with respect to $x$ , the derivative of $- 6 x π$ with respect to $x$ is $- 6 π \frac{d}{d x} [x]$ .

$lim_{x \to π} \frac{4 x - 6 π \frac{d}{d x} [x] + \frac{d}{d x} [4 π^{2}]}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.4.2

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

$lim_{x \to π} \frac{4 x - 6 π \cdot 1 + \frac{d}{d x} [4 π^{2}]}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.4.3

Multiply $- 6$ by $1$ .

$lim_{x \to π} \frac{4 x - 6 π + \frac{d}{d x} [4 π^{2}]}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.5

Since $4 π^{2}$ is constant with respect to $x$ , the derivative of $4 π^{2}$ with respect to $x$ is $0$ .

$lim_{x \to π} \frac{4 x - 6 π + 0}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.6

Add $4 x - 6 π$ and $0$ .

$lim_{x \to π} \frac{4 x - 6 π}{\frac{d}{d x} [x^{2} - π^{2}]}$

Step 1.3.7

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

$lim_{x \to π} \frac{4 x - 6 π}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- π^{2}]}$

Step 1.3.8

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

$lim_{x \to π} \frac{4 x - 6 π}{2 x + \frac{d}{d x} [- π^{2}]}$

Step 1.3.9

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

$lim_{x \to π} \frac{4 x - 6 π}{2 x + 0}$

Step 1.3.10

Add $2 x$ and $0$ .

$lim_{x \to π} \frac{4 x - 6 π}{2 x}$

Step 1.4

Cancel the common factor of $4 x - 6 π$ and $2$ .

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Step 1.4.1

Factor $2$ out of $4 x$ .

$lim_{x \to π} \frac{2 (2 x) - 6 π}{2 x}$

Step 1.4.2

Factor $2$ out of $- 6 π$ .

$lim_{x \to π} \frac{2 (2 x) + 2 (- 3 π)}{2 x}$

Step 1.4.3

Factor $2$ out of $2 (2 x) + 2 (- 3 π)$ .

$lim_{x \to π} \frac{2 (2 x - 3 π)}{2 x}$

Step 1.4.4

Cancel the common factors.

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Step 1.4.4.1

Factor $2$ out of $2 x$ .

$lim_{x \to π} \frac{2 (2 x - 3 π)}{2 (x)}$

Step 1.4.4.2

Cancel the common factor.

$lim_{x \to π} \frac{2 (2 x - 3 π)}{2 x}$

Step 1.4.4.3

Rewrite the expression.

$lim_{x \to π} \frac{2 x - 3 π}{x}$

Step 2

Evaluate the limit.

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Step 2.1

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

$\frac{lim_{x \to π} 2 x - 3 π}{lim_{x \to π} x}$

Step 2.2

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

$\frac{lim_{x \to π} 2 x - lim_{x \to π} 3 π}{lim_{x \to π} x}$

Step 2.3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to π} x - lim_{x \to π} 3 π}{lim_{x \to π} x}$

Step 2.4

Evaluate the limit of $3 π$ which is constant as $x$ approaches $π$ .

$\frac{2 lim_{x \to π} x - 3 π}{lim_{x \to π} x}$

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 $x$ .

$\frac{2 π - 3 π}{lim_{x \to π} x}$

Step 3.2

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

$\frac{2 π - 3 π}{π}$

Step 4

Simplify the answer.

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Step 4.1

Subtract $3 π$ from $2 π$ .

$\frac{- π}{π}$

Step 4.2

Cancel the common factor of $π$ .

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Step 4.2.1

Cancel the common factor.

$\frac{- π}{π}$

Step 4.2.2

Divide $- 1$ by $1$ .

$- 1$