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.

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.

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.

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.

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

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

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.

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.

Anna,

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

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

no its a 2.

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.

To evaluate the limit analytically without using a table or a graph, you can follow these steps:

1. Simplify the expression:
Start by simplifying the given expression: (sqrt(2-x) - 2)/x.

2. Factor out a common factor:
Let's factor out a common factor of (√(2-x) - 2). This will help us simplify the expression further.

3. Rationalize the denominator:
The denominator x can be rationalized by multiplying the numerator and denominator by the conjugate of the denominator, which in this case is (√(2-x) + 2). By multiplying, we eliminate the square root in the denominator.

4. Simplify the expression further:
After rationalizing the denominator, simplify the expression by expanding the numerator.

5. Evaluate the limit:
Now that the expression is simplified, you can evaluate the limit as x approaches 0. This can be done by directly substituting 0 into the expression and calculating the resulting value.

If the resulting value is a finite number, then the limit exists. If it approaches positive infinity (∞) or negative infinity (-∞), the limit does not exist.

In the case of your specific problem, the steps mentioned above can be applied to evaluate the limit of (sqrt(2-x)-2)/x as x approaches 0.