Calculus Examples

$\frac{lim_{x \to 1} x^{100} - x^{2} + x - 1}{x^{10} - 1}$

Step 1

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

$\frac{lim_{x \to 1} x^{100} - lim_{x \to 1} x^{2} + lim_{x \to 1} x - lim_{x \to 1} 1}{x^{10} - 1}$

Step 2

Move the exponent $100$ from $x^{100}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{100} - lim_{x \to 1} x^{2} + lim_{x \to 1} x - lim_{x \to 1} 1}{x^{10} - 1}$

Step 3

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

$\frac{{(lim_{x \to 1} x)}^{100} - {(lim_{x \to 1} x)}^{2} + lim_{x \to 1} x - lim_{x \to 1} 1}{x^{10} - 1}$

Step 4

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

$\frac{{(lim_{x \to 1} x)}^{100} - {(lim_{x \to 1} x)}^{2} + lim_{x \to 1} x - 1 \cdot 1}{x^{10} - 1}$

Step 5

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

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Step 5.1

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

$\frac{1^{100} - {(lim_{x \to 1} x)}^{2} + lim_{x \to 1} x - 1 \cdot 1}{x^{10} - 1}$

Step 5.2

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

$\frac{1^{100} - 1^{2} + lim_{x \to 1} x - 1 \cdot 1}{x^{10} - 1}$

Step 5.3

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

$\frac{1^{100} - 1^{2} + 1 - 1 \cdot 1}{x^{10} - 1}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

One to any power is one.

$\frac{1 - 1^{2} + 1 - 1 \cdot 1}{x^{10} - 1}$

Step 6.1.2

One to any power is one.

$\frac{1 - 1 \cdot 1 + 1 - 1 \cdot 1}{x^{10} - 1}$

Step 6.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - 1 + 1 - 1 \cdot 1}{x^{10} - 1}$

Step 6.1.4

Multiply $- 1$ by $1$ .

$\frac{1 - 1 + 1 - 1}{x^{10} - 1}$

Step 6.1.5

Subtract $1$ from $1$ .

$\frac{0 + 1 - 1}{x^{10} - 1}$

Step 6.1.6

Add $0$ and $1$ .

$\frac{1 - 1}{x^{10} - 1}$

Step 6.1.7

Subtract $1$ from $1$ .

$\frac{0}{x^{10} - 1}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Rewrite $x^{10}$ as ${(x^{5})}^{2}$ .

$\frac{0}{{(x^{5})}^{2} - 1}$

Step 6.2.2

Rewrite $1$ as $1^{2}$ .

$\frac{0}{{(x^{5})}^{2} - 1^{2}}$

Step 6.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{5}$ and $b = 1$ .

$\frac{0}{(x^{5} + 1) (x^{5} - 1)}$

Step 6.3

Divide $0$ by $(x^{5} + 1) (x^{5} - 1)$ .

$0$