Math Help - Sandwich Theorem

1. Sandwich Theorem

If sqrt(5-2x^2) <= f(x) <= sqrt(5-x^2) for -1 <= x <= 1, find limit f(x) as x goes to 0.

What exactly is the first step in doing these types of problems? This theorem is the only one that I have trouble with as I missed the day of class when we were discussing it.

<= is less than or equal to. I guess I will have to find the latex codes for these problems.

2. In simple words, in the left side what's the limit? in the right one, what's the limit? are those equal? if so, then the limit of the squeezed function is the same.

3. Originally Posted by VitaX
If sqrt(5-2x^2) <= f(x) <= sqrt(5-x^2) for -1 <= x <= 1, find limit f(x) as x goes to 0.

What exactly is the first step in doing these types of problems? This theorem is the only one that I have trouble with as I missed the day of class when we were discussing it.

<= is less than or equal to. I guess I will have to find the latex codes for these problems.
yup!

you do have to check if the left side limit is equal to the right side. if they are the same, limit of f(x) will also be the same.

we can see that,

limit of sqrt(5-2x^2) = sqrt 5 when x tends to 0

limit of sqrt(5-x^2) = sqrt 5 when x tends to 0

Hence from here, we can conclude that

limit of f(x) = sqrt 5 when x tends to 0

4. So basically it is just simple substitution and if both the left and right side are equal, then f(x) is equal as well. What if the left and right were different values. What would be the next step?

5. Originally Posted by VitaX
So basically it is just simple substitution and if both the left and right side are equal, then f(x) is equal as well. What if the left and right were different values. What would be the next step?
if the left and right value are not equal, we cannot use the sandwich theorem to find the limit for f(x). if we know the equation of the f(x), other methods can be used to find the limits. by using one sided limits.