Calculus Examples

$lim_{x \to a} \frac{x^{n} - a^{n}}{x - a}$

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.

$\frac{lim_{x \to a} x^{n} - a^{n}}{lim_{x \to a} x - a}$

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 $a$ .

$\frac{lim_{x \to a} x^{n} - lim_{x \to a} a^{n}}{lim_{x \to a} x - a}$

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

Evaluate the limit of $a^{n}$ which is constant as $x$ approaches $a$ .

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

Step 1.1.2.2

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

$\frac{a^{n} - a^{n}}{lim_{x \to a} x - a}$

Step 1.1.2.3

Subtract $a^{n}$ from $a^{n}$ .

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

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 $a$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.2

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

$\frac{0}{a - a}$

Step 1.1.3.3

Subtract $a$ from $a$ .

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

$lim_{x \to a} \frac{n x^{n - 1} + \frac{d}{d x} [- a^{n}]}{\frac{d}{d x} [x - a]}$

Step 1.3.4

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

$lim_{x \to a} \frac{n x^{n - 1} + 0}{\frac{d}{d x} [x - a]}$

Step 1.3.5

Simplify.

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Step 1.3.5.1

Add $n x^{n - 1}$ and $0$ .

$lim_{x \to a} \frac{n x^{n - 1}}{\frac{d}{d x} [x - a]}$

Step 1.3.5.2

Reorder the factors of $n x^{n - 1}$ .

$lim_{x \to a} \frac{x^{n - 1} n}{\frac{d}{d x} [x - a]}$

Step 1.3.5.3

Reorder factors in $x^{n - 1} n$ .

$lim_{x \to a} \frac{n x^{n - 1}}{\frac{d}{d x} [x - a]}$

Step 1.3.6

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

$lim_{x \to a} \frac{n x^{n - 1}}{\frac{d}{d x} [x] + \frac{d}{d x} [- a]}$

Step 1.3.7

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

$lim_{x \to a} \frac{n x^{n - 1}}{1 + \frac{d}{d x} [- a]}$

Step 1.3.8

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

$lim_{x \to a} \frac{n x^{n - 1}}{1 + 0}$

Step 1.3.9

Add $1$ and $0$ .

$lim_{x \to a} \frac{n x^{n - 1}}{1}$

Step 1.4

Divide $n x^{n - 1}$ by $1$ .

$lim_{x \to a} n x^{n - 1}$

Step 2

Evaluate the limit.

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Step 2.1

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

$n lim_{x \to a} x^{n - 1}$

Step 2.2

Move the exponent $n - 1$ from $x^{n - 1}$ outside the limit using the Limits Power Rule.

$n {(lim_{x \to a} x)}^{n - 1}$

Step 3

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

$n a^{n - 1}$