Mathematics, 19.08.2020 21:01 sarinaneedshelp01

Silly and/or spam answers will not be tolerated Evaluate the limit: $\lim_{x \to -6} \frac{\sqrt{10 - x}-4 }{x + 6}$
Please explain how to rationalize and solve for the limit.

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Answer from: dejaproctor17

-1/8

Step-by-step explanation:

lim x approaches -6 (sqrt( 10-x) -4) / (x+6)

Rationalize

(sqrt( 10-x) -4) (sqrt( 10-x) +4)

(x+6) (sqrt( 10-x) +4)

We know ( a-b) (a+b) = a^2 -b^2

a= ( sqrt(10-x) b = 4

(10-x) -16

(x+6) (sqrt( 10-x) +4)

-6-x

(x+6) (sqrt( 10-x) +4)

Factor out -1 from the numerator

-1( x+6)

(x+6) (sqrt( 10-x) +4)

Cancel x+6 from the numerator and denominator

-1

(sqrt( 10-x) +4)

Now take the limit

lim x approaches -6 -1/ (sqrt( 10-x) +4)

-1/ (sqrt( 10- -6) +4)

-1/ (sqrt(16) +4)

-1 /( 4+4)

-1/8

Answer from: jeffro198004

$\displaystyle \lim_{x \to -6}\frac{\sqrt{10-x}-4}{x+6} =-\frac{1}{8}$

Step-by-step explanation:

We want to evaluate the limit:

$\displaystyle \lim_{x \to -6}\frac{\sqrt{10-x}-4}{x+6} \\$

When attempting to evaluate a limit, we should always try direct substitution. This yields:

$\displaystyle \begin{aligned} &\Rightarrow \frac{\sqrt{10-(-6)}-4}{(-6)+6} \\ \\ &=\frac{\sqrt{16}-4}{-6+6}\\ \\ &=\frac{4-4}{-6+6}\\ \\ &=\underbrace{\frac{0}{0}}_{\text{Indeterminate}} \end{aligned}$

Since the result is an indeterminate form, we can try simplifying the limit.

Let's cancel the square root in the numerator. We can use the difference of two squares. Recall that:

(a-b)(a+b)=a^2-b^2

The expression in the numerator is:

$\sqrt{10-x}-4$

Therefore, to cancel it out, we will multiply it by:

$\sqrt{10-x}+4$

Multiply. This yields:

$=\displaystyle \lim_{x \to -6}\frac{\sqrt{10-x}-4}{x+6}\cdot\frac{\sqrt{10-x}+4}{\sqrt{10-x}+4} \\$

Simplify:

$\displaystyle \begin{aligned} &= \lim_{x \to -6}\frac{(\sqrt{10-x})^2-(4)^2}{(x+6)(\sqrt{10-x}+4)}\\ \\ &=\lim_{x\to-6}\frac{(10-x)-(16)}{(x+6)(\sqrt{10-x}+4)}\\ \\ &=\lim_{x \to \ -6}\frac{-x-6}{x+6(\sqrt{10-x}+4)}\end{aligned}$

Factor:

$\displaystyle \lim_{x \to \ -6}\frac{-(x+6)}{(x+6)(\sqrt{10-x}+4)}$

Cancel:

$\displaystyle \lim_{x \to \ -6}-\frac{1}{\sqrt{10-x}+4}$

Now, we can attempt direct substitution again. Thus:

$\displaystyle \begin{aligned} &\Rightarrow -\frac{1}{(\sqrt{10-(-6)}+4)}\\ \\ &=-\frac{1}{\sqrt{16+4}}\\ \\ &=-\frac{1}{(4+4)} \\ \\ &=-\frac{1}{8}\end{aligned}$

Therefore:

$\displaystyle \lim_{x \to -6}\frac{\sqrt{10-x}-4}{x+6} =-\frac{1}{8}$

Answer from: Quest

answer: evaluate the objective function at each vertex to determine which - and -values, if any, maximize or minimize the function. example: find the minimum value and maximum value of the objective function f ( x , y ) = 4 x + 5 y , subject to the following constraints

Answer from: Quest

step-by-step explanation:

answers available in this download

(pearson%2c%25202012)%2520bbs.pdf& usg=aovvaw1obmtxnfghfgxhx0urh_je

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