Calculus Examples

$lim_{x \to 6} (\sqrt{6 x} + 3 x - \frac{1}{x}) (x^{2} - 4)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $6$ .

$lim_{x \to 6} \sqrt{6 x} + 3 x - \frac{1}{x} \cdot lim_{x \to 6} x^{2} - 4$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$(lim_{x \to 6} \sqrt{6 x} + lim_{x \to 6} 3 x - lim_{x \to 6} \frac{1}{x}) \cdot lim_{x \to 6} x^{2} - 4$

Step 3

Move the limit under the radical sign.

$(\sqrt{lim_{x \to 6} 6 x} + lim_{x \to 6} 3 x - lim_{x \to 6} \frac{1}{x}) \cdot lim_{x \to 6} x^{2} - 4$

Step 4

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

$(\sqrt{6 lim_{x \to 6} x} + lim_{x \to 6} 3 x - lim_{x \to 6} \frac{1}{x}) \cdot lim_{x \to 6} x^{2} - 4$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$(\sqrt{6 lim_{x \to 6} x} + 3 lim_{x \to 6} x - lim_{x \to 6} \frac{1}{x}) \cdot lim_{x \to 6} x^{2} - 4$

Step 6

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $6$ .

$(\sqrt{6 lim_{x \to 6} x} + 3 lim_{x \to 6} x - \frac{lim_{x \to 6} 1}{lim_{x \to 6} x}) \cdot lim_{x \to 6} x^{2} - 4$

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $6$ .

$(\sqrt{6 lim_{x \to 6} x} + 3 lim_{x \to 6} x - \frac{1}{lim_{x \to 6} x}) \cdot lim_{x \to 6} x^{2} - 4$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $6$ .

$(\sqrt{6 lim_{x \to 6} x} + 3 lim_{x \to 6} x - \frac{1}{lim_{x \to 6} x}) \cdot (lim_{x \to 6} x^{2} - lim_{x \to 6} 4)$

Step 9

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

$(\sqrt{6 lim_{x \to 6} x} + 3 lim_{x \to 6} x - \frac{1}{lim_{x \to 6} x}) \cdot ({(lim_{x \to 6} x)}^{2} - lim_{x \to 6} 4)$

Step 10

Evaluate the limit of $4$ which is constant as $x$ approaches $6$ .

$(\sqrt{6 lim_{x \to 6} x} + 3 lim_{x \to 6} x - \frac{1}{lim_{x \to 6} x}) \cdot ({(lim_{x \to 6} x)}^{2} - 1 \cdot 4)$

Step 11

Evaluate the limits by plugging in $6$ for all occurrences of $x$ .

Tap for more steps...

Step 11.1

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$(\sqrt{6 \cdot 6} + 3 lim_{x \to 6} x - \frac{1}{lim_{x \to 6} x}) \cdot ({(lim_{x \to 6} x)}^{2} - 1 \cdot 4)$

Step 11.2

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$(\sqrt{6 \cdot 6} + 3 \cdot 6 - \frac{1}{lim_{x \to 6} x}) \cdot ({(lim_{x \to 6} x)}^{2} - 1 \cdot 4)$

Step 11.3

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$(\sqrt{6 \cdot 6} + 3 \cdot 6 - \frac{1}{6}) \cdot ({(lim_{x \to 6} x)}^{2} - 1 \cdot 4)$

Step 11.4

Evaluate the limit of $x$ by plugging in $6$ for $x$ .

$(\sqrt{6 \cdot 6} + 3 \cdot 6 - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12

Simplify the answer.

Tap for more steps...

Step 12.1

Simplify each term.

Tap for more steps...

Step 12.1.1

Multiply $6$ by $6$ .

$(\sqrt{36} + 3 \cdot 6 - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12.1.2

Rewrite $36$ as $6^{2}$ .

$(\sqrt{6^{2}} + 3 \cdot 6 - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12.1.3

Pull terms out from under the radical, assuming positive real numbers.

$(6 + 3 \cdot 6 - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12.1.4

Multiply $3$ by $6$ .

$(6 + 18 - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12.2

Find the common denominator.

Tap for more steps...

Step 12.2.1

Write $6$ as a fraction with denominator $1$ .

$(\frac{6}{1} + 18 - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12.2.2

Multiply $\frac{6}{1}$ by $\frac{6}{6}$ .

$(\frac{6}{1} \cdot \frac{6}{6} + 18 - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12.2.3

Multiply $\frac{6}{1}$ by $\frac{6}{6}$ .

$(\frac{6 \cdot 6}{6} + 18 - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12.2.4

Write $18$ as a fraction with denominator $1$ .

$(\frac{6 \cdot 6}{6} + \frac{18}{1} - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12.2.5

Multiply $\frac{18}{1}$ by $\frac{6}{6}$ .

$(\frac{6 \cdot 6}{6} + \frac{18}{1} \cdot \frac{6}{6} - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12.2.6

Multiply $\frac{18}{1}$ by $\frac{6}{6}$ .

$(\frac{6 \cdot 6}{6} + \frac{18 \cdot 6}{6} - \frac{1}{6}) \cdot (6^{2} - 1 \cdot 4)$

Step 12.3

Combine the numerators over the common denominator.

$\frac{6 \cdot 6 + 18 \cdot 6 - 1}{6} \cdot (6^{2} - 1 \cdot 4)$

Step 12.4

Simplify each term.

Tap for more steps...

Step 12.4.1

Multiply $6$ by $6$ .

$\frac{36 + 18 \cdot 6 - 1}{6} \cdot (6^{2} - 1 \cdot 4)$

Step 12.4.2

Multiply $18$ by $6$ .

$\frac{36 + 108 - 1}{6} \cdot (6^{2} - 1 \cdot 4)$

Step 12.5

Add $36$ and $108$ .

$\frac{144 - 1}{6} \cdot (6^{2} - 1 \cdot 4)$

Step 12.6

Subtract $1$ from $144$ .

$\frac{143}{6} \cdot (6^{2} - 1 \cdot 4)$

Step 12.7

Simplify each term.

Tap for more steps...

Step 12.7.1

Raise $6$ to the power of $2$ .

$\frac{143}{6} \cdot (36 - 1 \cdot 4)$

Step 12.7.2

Multiply $- 1$ by $4$ .

$\frac{143}{6} \cdot (36 - 4)$

Step 12.8

Subtract $4$ from $36$ .

$\frac{143}{6} \cdot 32$

Step 12.9

Cancel the common factor of $2$ .

Tap for more steps...

Step 12.9.1

Factor $2$ out of $6$ .

$\frac{143}{2 (3)} \cdot 32$

Step 12.9.2

Factor $2$ out of $32$ .

$\frac{143}{2 \cdot 3} \cdot (2 \cdot 16)$

Step 12.9.3

Cancel the common factor.

$\frac{143}{2 \cdot 3} \cdot (2 \cdot 16)$

Step 12.9.4

Rewrite the expression.

$\frac{143}{3} \cdot 16$

Step 12.10

Combine $\frac{143}{3}$ and $16$ .

$\frac{143 \cdot 16}{3}$

Step 12.11

Multiply $143$ by $16$ .

$\frac{2288}{3}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{2288}{3}$

Decimal Form:

$762.\overline{6}$