Calculus Examples

$lim_{x \to 11^{+}} \frac{110}{6 x^{2} - 102 x + 396} - \frac{2 x}{6 x - 66}$

Step 1

Reduce.

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Step 1.1

Cancel the common factor of $110$ and $6 x^{2} - 102 x + 396$ .

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Step 1.1.1

Factor $2$ out of $110$ .

$lim_{x \to 11^{+}} \frac{2 (55)}{6 x^{2} - 102 x + 396} - \frac{2 x}{6 x - 66}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $6 x^{2}$ .

$lim_{x \to 11^{+}} \frac{2 (55)}{2 (3 x^{2}) - 102 x + 396} - \frac{2 x}{6 x - 66}$

Step 1.1.2.2

Factor $2$ out of $- 102 x$ .

$lim_{x \to 11^{+}} \frac{2 (55)}{2 (3 x^{2}) + 2 (- 51 x) + 396} - \frac{2 x}{6 x - 66}$

Step 1.1.2.3

Factor $2$ out of $2 (3 x^{2}) + 2 (- 51 x)$ .

$lim_{x \to 11^{+}} \frac{2 (55)}{2 (3 x^{2} - 51 x) + 396} - \frac{2 x}{6 x - 66}$

Step 1.1.2.4

Factor $2$ out of $396$ .

$lim_{x \to 11^{+}} \frac{2 \cdot 55}{2 (3 x^{2} - 51 x) + 2 \cdot 198} - \frac{2 x}{6 x - 66}$

Step 1.1.2.5

Factor $2$ out of $2 (3 x^{2} - 51 x) + 2 (198)$ .

$lim_{x \to 11^{+}} \frac{2 (55)}{2 (3 x^{2} - 51 x + 198)} - \frac{2 x}{6 x - 66}$

Step 1.1.2.6

Cancel the common factor.

$lim_{x \to 11^{+}} \frac{2 \cdot 55}{2 (3 x^{2} - 51 x + 198)} - \frac{2 x}{6 x - 66}$

Step 1.1.2.7

Rewrite the expression.

$lim_{x \to 11^{+}} \frac{55}{3 x^{2} - 51 x + 198} - \frac{2 x}{6 x - 66}$

Step 1.2

Cancel the common factor of $2$ and $6 x - 66$ .

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Step 1.2.1

Factor $2$ out of $2 x$ .

$lim_{x \to 11^{+}} \frac{55}{3 x^{2} - 51 x + 198} - \frac{2 (x)}{6 x - 66}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $2$ out of $6 x$ .

$lim_{x \to 11^{+}} \frac{55}{3 x^{2} - 51 x + 198} - \frac{2 (x)}{2 (3 x) - 66}$

Step 1.2.2.2

Factor $2$ out of $- 66$ .

$lim_{x \to 11^{+}} \frac{55}{3 x^{2} - 51 x + 198} - \frac{2 x}{2 (3 x) + 2 \cdot - 33}$

Step 1.2.2.3

Factor $2$ out of $2 (3 x) + 2 (- 33)$ .

$lim_{x \to 11^{+}} \frac{55}{3 x^{2} - 51 x + 198} - \frac{2 (x)}{2 (3 x - 33)}$

Step 1.2.2.4

Cancel the common factor.

$lim_{x \to 11^{+}} \frac{55}{3 x^{2} - 51 x + 198} - \frac{2 x}{2 (3 x - 33)}$

Step 1.2.2.5

Rewrite the expression.

$lim_{x \to 11^{+}} \frac{55}{3 x^{2} - 51 x + 198} - \frac{x}{3 x - 33}$

Step 2

Make a table to show the behavior of the function $\frac{55}{3 x^{2} - 51 x + 198} - \frac{x}{3 x - 33}$ as $x$ approaches $11$ from the right.

$\begin{array}{c} x & \frac{55}{3 x^{2} - 51 x + 198} - \frac{x}{3 x - 33} \\ 11.1 & - 1.05228758 \\ 11.01 & - 1.06520292 \\ 11.001 & - 1.06652002 \\ 11.0001 & - 1.066652 \end{array}$

Step 3

As the $x$ values approach $11$ , the function values approach $- 1.067$ . Thus, the limit of $\frac{55}{3 x^{2} - 51 x + 198} - \frac{x}{3 x - 33}$ as $x$ approaches $11$ from the right is $- 1.067$ .

$- 1.067$