Calculus Examples

$\frac{lim_{x \to 7} x + \sqrt{x - 6}}{x^{2} - 2 x + 1}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$\frac{lim_{x \to 7} x + lim_{x \to 7} \sqrt{x - 6}}{x^{2} - 2 x + 1}$

Step 2

Move the limit under the radical sign.

$\frac{lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 6}}{x^{2} - 2 x + 1}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$\frac{lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - lim_{x \to 7} 6}}{x^{2} - 2 x + 1}$

Step 4

Evaluate the limit of $6$ which is constant as $x$ approaches $7$ .

$\frac{lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}}{x^{2} - 2 x + 1}$

Step 5

Evaluate the limits by plugging in $7$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\frac{7 + \sqrt{lim_{x \to 7} x - 1 \cdot 6}}{x^{2} - 2 x + 1}$

Step 5.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\frac{7 + \sqrt{7 - 1 \cdot 6}}{x^{2} - 2 x + 1}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- 1$ by $6$ .

$\frac{7 + \sqrt{7 - 6}}{x^{2} - 2 x + 1}$

Step 6.1.2

Subtract $6$ from $7$ .

$\frac{7 + \sqrt{1}}{x^{2} - 2 x + 1}$

Step 6.1.3

Any root of $1$ is $1$ .

$\frac{7 + 1}{x^{2} - 2 x + 1}$

Step 6.1.4

Add $7$ and $1$ .

$\frac{8}{x^{2} - 2 x + 1}$

Step 6.2

Factor using the perfect square rule.

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Step 6.2.1

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

$\frac{8}{x^{2} - 2 x + 1^{2}}$

Step 6.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 6.2.3

Rewrite the polynomial.

$\frac{8}{x^{2} - 2 \cdot x \cdot 1 + 1^{2}}$

Step 6.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$\frac{8}{{(x - 1)}^{2}}$