Question

Our professor wants us to evaluate the limits analytically without using a table or a graph, and if it doesn't exist we must describe the behavor near the limit point. I'm not sure how to evaluate each side of a limit separately without looking at a table or graph. Here is one of the problems:

The limit of (sqrt(2-x)-2)/x as x approaches 0.

Answer 1

Didn't I show you this one? Rationalize the numerator;

lim (sqrt(2-x)-2)(sqrt(2-x)+2)/(x(sqrt(2-x)+2)

which equals
lim ((2-x-4)/(x(sqrt(x-2)+2)

which equals
lim (x-4)/(x(sqrt(x-2)+2)

as x approaches zero
-4/zero which does not exist.

Answer 2

But now if it does not exist we have to go back and look at the limit as the function approaches from the left and from the right and describe the behavoir of the function for example if it approaches infinity or negative infinity. I don't know how to do that analytically without using a graph or a chart.

Answer 3

I'm not sure if you learned about L'hospital rule
http://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx

(sqrt(2-x)-2)/x

rewrite:
[(2-x)^(-1/2)-2]/x

perform L'hospital's by taking the derivative of numerator and denominator individually:
[(1/2)(2-x)^(-1/2)]/1

now take the limit.

Answer 4

opps, forgot about chain rule! be sure to take the derivative of the inside (2-x)!

[(-1/2)(2-x)^(-1/2)]/1

Answer 5

look at the left. Let x approach from the left (x is negative)
lim (2-x)-2)/x means the denominator is negative, the numberator is positive, so the limit is negative.
Now, from the right, x is positive
lim (2-x)-2)/x numerator is negative, denominator is positive, lim is negative.

Answer 6

We don't start derivatives until the next chapter. I'm just don't understand how you can determine each side of a limit separately without using a graph or chart.

Answer 7

Anna,
can you double check if the question is not:
"The limit of (sqrt(<b>4</b>-x)-2)/x as x approaches 0"

The answer would be -1/4 using Mr. Pursley's approach.

Answer 8

no its a 2.

Answer 9

Way back in the stone age when I taught math, I gave
my students the following procedure when doing limits.
Subthe approach value in the given expression

1. If you get a real number as an answer, that's it.
That is your answer, go on to the next question.
2. If you get a/0 , where a ≠ 0, then the limit is undefined.
3. If you get 0/0, you got yourself a real limit question. Try rationalizing, factoring, substitution or other clever math procedures.

in your case, unless there is a typo like MathMate suspects, you would get (√2 - 2)/0 which is undefined.