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

step-by-step explanation:

256/6=42

Answer from: Quest

introduction, methodology, results, discussion

step-by-step explanation:

an observational study is one which consists of actual observation to record data and then analyse the data.

there are a number of factors that should be analyzed before starting an observational study.

the main four factors are:

i) introduction: this should be analysed to check whether the purpose of the study is correctly established and the data recorded are appropriate for the study

methodology: the method of collecting the data, such as whether randomly selected, and whether sample size matches with the population to be studied, whether questions asked are easy to answer by common men, etc.

results: then results such as summary of data with comments should be studied thoroughly

discussion: the last is the discussion, this can be either by management people or persons who are all interested , etc.

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