Calculus Examples

$lim_{x \to 9} \frac{x^{9} - x^{x}}{x - 9}$

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.

$\frac{lim_{x \to 9} x^{9} - x^{x}}{lim_{x \to 9} x - 9}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $9$ .

$\frac{lim_{x \to 9} x^{9} - lim_{x \to 9} x^{x}}{lim_{x \to 9} x - 9}$

Step 1.1.2.1.2

Move the exponent $9$ from $x^{9}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 9} x)}^{9} - lim_{x \to 9} x^{x}}{lim_{x \to 9} x - 9}$

Step 1.1.2.2

Use the properties of logarithms to simplify the limit.

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Step 1.1.2.2.1

Rewrite $x^{x}$ as $e^{\ln (x^{x})}$ .

$\frac{{(lim_{x \to 9} x)}^{9} - lim_{x \to 9} e^{\ln (x^{x})}}{lim_{x \to 9} x - 9}$

Step 1.1.2.2.2

Expand $\ln (x^{x})$ by moving $x$ outside the logarithm.

$\frac{{(lim_{x \to 9} x)}^{9} - lim_{x \to 9} e^{x \ln (x)}}{lim_{x \to 9} x - 9}$

Step 1.1.2.3

Evaluate the limit.

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Step 1.1.2.3.1

Move the limit into the exponent.

$\frac{{(lim_{x \to 9} x)}^{9} - e^{lim_{x \to 9} x \ln (x)}}{lim_{x \to 9} x - 9}$

Step 1.1.2.3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $9$ .

$\frac{{(lim_{x \to 9} x)}^{9} - e^{lim_{x \to 9} x \cdot lim_{x \to 9} \ln (x)}}{lim_{x \to 9} x - 9}$

Step 1.1.2.3.3

Move the limit inside the logarithm.

$\frac{{(lim_{x \to 9} x)}^{9} - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)}}{lim_{x \to 9} x - 9}$

Step 1.1.2.4

Evaluate the limits by plugging in $9$ for all occurrences of $x$ .

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Step 1.1.2.4.1

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{9^{9} - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)}}{lim_{x \to 9} x - 9}$

Step 1.1.2.4.2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{9^{9} - e^{9 \cdot \ln (lim_{x \to 9} x)}}{lim_{x \to 9} x - 9}$

Step 1.1.2.4.3

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{9^{9} - e^{9 \cdot \ln (9)}}{lim_{x \to 9} x - 9}$

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

Raise $9$ to the power of $9$ .

$\frac{387420489 - e^{9 \cdot \ln (9)}}{lim_{x \to 9} x - 9}$

Step 1.1.2.5.1.2

Simplify $9 \cdot \ln (9)$ by moving $9$ inside the logarithm.

$\frac{387420489 - e^{\ln (9^{9})}}{lim_{x \to 9} x - 9}$

Step 1.1.2.5.1.3

Exponentiation and log are inverse functions.

$\frac{387420489 - 9^{9}}{lim_{x \to 9} x - 9}$

Step 1.1.2.5.1.4

Raise $9$ to the power of $9$ .

$\frac{387420489 - 1 \cdot 387420489}{lim_{x \to 9} x - 9}$

Step 1.1.2.5.1.5

Multiply $- 1$ by $387420489$ .

$\frac{387420489 - 387420489}{lim_{x \to 9} x - 9}$

Step 1.1.2.5.2

Subtract $387420489$ from $387420489$ .

$\frac{0}{lim_{x \to 9} x - 9}$

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 $9$ .

$\frac{0}{lim_{x \to 9} x - lim_{x \to 9} 9}$

Step 1.1.3.1.2

Evaluate the limit of $9$ which is constant as $x$ approaches $9$ .

$\frac{0}{lim_{x \to 9} x - 1 \cdot 9}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{0}{9 - 1 \cdot 9}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $9$ .

$\frac{0}{9 - 9}$

Step 1.1.3.3.2

Subtract $9$ from $9$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 9} \frac{x^{9} - x^{x}}{x - 9} = lim_{x \to 9} \frac{\frac{d}{d x} [x^{9} - x^{x}]}{\frac{d}{d x} [x - 9]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 9} \frac{\frac{d}{d x} [x^{9} - x^{x}]}{\frac{d}{d x} [x - 9]}$

Step 1.3.2

By the Sum Rule, the derivative of $x^{9} - x^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{9}] + \frac{d}{d x} [- x^{x}]$ .

$lim_{x \to 9} \frac{\frac{d}{d x} [x^{9}] + \frac{d}{d x} [- x^{x}]}{\frac{d}{d x} [x - 9]}$

Step 1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$lim_{x \to 9} \frac{9 x^{8} + \frac{d}{d x} [- x^{x}]}{\frac{d}{d x} [x - 9]}$

Step 1.3.4

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

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Step 1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{x}$ with respect to $x$ is $- \frac{d}{d x} [x^{x}]$ .

$lim_{x \to 9} \frac{9 x^{8} - \frac{d}{d x} [x^{x}]}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.2

Use the properties of logarithms to simplify the differentiation.

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Step 1.3.4.2.1

Rewrite $x^{x}$ as $e^{\ln (x^{x})}$ .

$lim_{x \to 9} \frac{9 x^{8} - \frac{d}{d x} [e^{\ln (x^{x})}]}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.2.2

Expand $\ln (x^{x})$ by moving $x$ outside the logarithm.

$lim_{x \to 9} \frac{9 x^{8} - \frac{d}{d x} [e^{x \ln (x)}]}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.3

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

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Step 1.3.4.3.1

To apply the Chain Rule, set $u$ as $x \ln (x)$ .

$lim_{x \to 9} \frac{9 x^{8} - (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x \ln (x)])}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to 9} \frac{9 x^{8} - (e^{u} \frac{d}{d x} [x \ln (x)])}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.3.3

Replace all occurrences of $u$ with $x \ln (x)$ .

$lim_{x \to 9} \frac{9 x^{8} - (e^{x \ln (x)} \frac{d}{d x} [x \ln (x)])}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (x)$ .

$lim_{x \to 9} \frac{9 x^{8} - (e^{x \ln (x)} (x \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x]))}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.5

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$lim_{x \to 9} \frac{9 x^{8} - (e^{x \ln (x)} (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x]))}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.6

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

$lim_{x \to 9} \frac{9 x^{8} - (e^{x \ln (x)} (x \frac{1}{x} + \ln (x) \cdot 1))}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.7

Combine $x$ and $\frac{1}{x}$ .

$lim_{x \to 9} \frac{9 x^{8} - (e^{x \ln (x)} (\frac{x}{x} + \ln (x) \cdot 1))}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.8

Cancel the common factor of $x$ .

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Step 1.3.4.8.1

Cancel the common factor.

$lim_{x \to 9} \frac{9 x^{8} - (e^{x \ln (x)} (\frac{x}{x} + \ln (x) \cdot 1))}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.8.2

Rewrite the expression.

$lim_{x \to 9} \frac{9 x^{8} - (e^{x \ln (x)} (1 + \ln (x) \cdot 1))}{\frac{d}{d x} [x - 9]}$

Step 1.3.4.9

Multiply $\ln (x)$ by $1$ .

$lim_{x \to 9} \frac{9 x^{8} - e^{x \ln (x)} (1 + \ln (x))}{\frac{d}{d x} [x - 9]}$

Step 1.3.5

Simplify.

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Step 1.3.5.1

Apply the distributive property.

$lim_{x \to 9} \frac{9 x^{8} - e^{x \ln (x)} \cdot 1 - e^{x \ln (x)} \ln (x)}{\frac{d}{d x} [x - 9]}$

Step 1.3.5.2

Multiply $- 1$ by $1$ .

$lim_{x \to 9} \frac{9 x^{8} - e^{x \ln (x)} - e^{x \ln (x)} \ln (x)}{\frac{d}{d x} [x - 9]}$

Step 1.3.5.3

Reorder terms.

$lim_{x \to 9} \frac{9 x^{8} - e^{x \ln (x)} \ln (x) - e^{x \ln (x)}}{\frac{d}{d x} [x - 9]}$

Step 1.3.6

By the Sum Rule, the derivative of $x - 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 9]$ .

$lim_{x \to 9} \frac{9 x^{8} - e^{x \ln (x)} \ln (x) - e^{x \ln (x)}}{\frac{d}{d x} [x] + \frac{d}{d x} [- 9]}$

Step 1.3.7

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

$lim_{x \to 9} \frac{9 x^{8} - e^{x \ln (x)} \ln (x) - e^{x \ln (x)}}{1 + \frac{d}{d x} [- 9]}$

Step 1.3.8

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$lim_{x \to 9} \frac{9 x^{8} - e^{x \ln (x)} \ln (x) - e^{x \ln (x)}}{1 + 0}$

Step 1.3.9

Add $1$ and $0$ .

$lim_{x \to 9} \frac{9 x^{8} - e^{x \ln (x)} \ln (x) - e^{x \ln (x)}}{1}$

Step 1.4

Divide $9 x^{8} - e^{x \ln (x)} \ln (x) - e^{x \ln (x)}$ by $1$ .

$lim_{x \to 9} 9 x^{8} - e^{x \ln (x)} \ln (x) - e^{x \ln (x)}$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $9$ .

$lim_{x \to 9} 9 x^{8} - lim_{x \to 9} e^{x \ln (x)} \ln (x) - lim_{x \to 9} e^{x \ln (x)}$

Step 2.2

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$9 lim_{x \to 9} x^{8} - lim_{x \to 9} e^{x \ln (x)} \ln (x) - lim_{x \to 9} e^{x \ln (x)}$

Step 2.3

Move the exponent $8$ from $x^{8}$ outside the limit using the Limits Power Rule.

$9 {(lim_{x \to 9} x)}^{8} - lim_{x \to 9} e^{x \ln (x)} \ln (x) - lim_{x \to 9} e^{x \ln (x)}$

Step 2.4

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $9$ .

$9 {(lim_{x \to 9} x)}^{8} - lim_{x \to 9} e^{x \ln (x)} \cdot lim_{x \to 9} \ln (x) - lim_{x \to 9} e^{x \ln (x)}$

Step 2.5

Move the limit into the exponent.

$9 {(lim_{x \to 9} x)}^{8} - e^{lim_{x \to 9} x \ln (x)} \cdot lim_{x \to 9} \ln (x) - lim_{x \to 9} e^{x \ln (x)}$

Step 2.6

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $9$ .

$9 {(lim_{x \to 9} x)}^{8} - e^{lim_{x \to 9} x \cdot lim_{x \to 9} \ln (x)} \cdot lim_{x \to 9} \ln (x) - lim_{x \to 9} e^{x \ln (x)}$

Step 2.7

Move the limit inside the logarithm.

$9 {(lim_{x \to 9} x)}^{8} - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)} \cdot lim_{x \to 9} \ln (x) - lim_{x \to 9} e^{x \ln (x)}$

Step 2.8

Move the limit inside the logarithm.

$9 {(lim_{x \to 9} x)}^{8} - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)} \cdot \ln (lim_{x \to 9} x) - lim_{x \to 9} e^{x \ln (x)}$

Step 2.9

Move the limit into the exponent.

$9 {(lim_{x \to 9} x)}^{8} - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)} \cdot \ln (lim_{x \to 9} x) - e^{lim_{x \to 9} x \ln (x)}$

Step 2.10

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $9$ .

$9 {(lim_{x \to 9} x)}^{8} - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)} \cdot \ln (lim_{x \to 9} x) - e^{lim_{x \to 9} x \cdot lim_{x \to 9} \ln (x)}$

Step 2.11

Move the limit inside the logarithm.

$9 {(lim_{x \to 9} x)}^{8} - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)} \cdot \ln (lim_{x \to 9} x) - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)}$

Step 3

Evaluate the limits by plugging in $9$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$9 \cdot 9^{8} - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)} \cdot \ln (lim_{x \to 9} x) - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)}$

Step 3.2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$9 \cdot 9^{8} - e^{9 \cdot \ln (lim_{x \to 9} x)} \cdot \ln (lim_{x \to 9} x) - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)}$

Step 3.3

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$9 \cdot 9^{8} - e^{9 \cdot \ln (9)} \cdot \ln (lim_{x \to 9} x) - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)}$

Step 3.4

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$9 \cdot 9^{8} - e^{9 \cdot \ln (9)} \cdot \ln (9) - e^{lim_{x \to 9} x \cdot \ln (lim_{x \to 9} x)}$

Step 3.5

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$9 \cdot 9^{8} - e^{9 \cdot \ln (9)} \cdot \ln (9) - e^{9 \cdot \ln (lim_{x \to 9} x)}$

Step 3.6

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$9 \cdot 9^{8} - e^{9 \cdot \ln (9)} \cdot \ln (9) - e^{9 \cdot \ln (9)}$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $9$ by $9^{8}$ by adding the exponents.

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Step 4.1.1.1

Multiply $9$ by $9^{8}$ .

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Step 4.1.1.1.1

Raise $9$ to the power of $1$ .

$9^{1} \cdot 9^{8} - e^{9 \cdot \ln (9)} \cdot \ln (9) - e^{9 \cdot \ln (9)}$

Step 4.1.1.1.2

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

$9^{1 + 8} - e^{9 \cdot \ln (9)} \cdot \ln (9) - e^{9 \cdot \ln (9)}$

Step 4.1.1.2

Add $1$ and $8$ .

$9^{9} - e^{9 \cdot \ln (9)} \cdot \ln (9) - e^{9 \cdot \ln (9)}$

Step 4.1.2

Raise $9$ to the power of $9$ .

$387420489 - e^{9 \cdot \ln (9)} \cdot \ln (9) - e^{9 \cdot \ln (9)}$

Step 4.1.3

Simplify $9 \cdot \ln (9)$ by moving $9$ inside the logarithm.

$387420489 - e^{\ln (9^{9})} \cdot \ln (9) - e^{9 \cdot \ln (9)}$

Step 4.1.4

Exponentiation and log are inverse functions.

$387420489 - 9^{9} \cdot \ln (9) - e^{9 \cdot \ln (9)}$

Step 4.1.5

Raise $9$ to the power of $9$ .

$387420489 - 1 \cdot 387420489 \cdot \ln (9) - e^{9 \cdot \ln (9)}$

Step 4.1.6

Multiply $- 1$ by $387420489$ .

$387420489 - 387420489 \cdot \ln (9) - e^{9 \cdot \ln (9)}$

Step 4.1.7

Simplify $9 \cdot \ln (9)$ by moving $9$ inside the logarithm.

$387420489 - 387420489 \ln (9) - e^{\ln (9^{9})}$

Step 4.1.8

Exponentiation and log are inverse functions.

$387420489 - 387420489 \ln (9) - 9^{9}$

Step 4.1.9

Raise $9$ to the power of $9$ .

$387420489 - 387420489 \ln (9) - 1 \cdot 387420489$

Step 4.1.10

Multiply $- 1$ by $387420489$ .

$387420489 - 387420489 \ln (9) - 387420489$

Step 4.2

Subtract $387420489$ from $387420489$ .

$0 - 387420489 \ln (9)$

Step 4.3

Subtract $387420489 \ln (9)$ from $0$ .

$- 387420489 \ln (9)$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- 387420489 \ln (9)$

Decimal Form:

$- 851249820.194417$