Use continuity to evaluate the limit. lim x→16 20 + x 20 + x step 1 consider the intervals for which the numerator and the denominator are continuous. the numerator 20 + x is continuous on the interval the denominator 20 + x is continuous and nonzero on the interval
We are given the expression, .
Since, we have that,
The numerator and denominator is (20+x), which is a polynomial is continuous for all the real numbers i.e. for all x belonging to .
Thus, the limit exists and is given by,
= = = 1.
So, = 1.
You can stop right there, or you can try finding the exact value of .
Recall DeMoivre's theorem:
This means when , the imaginary part of the expansion of the left side will give you an expanded form of in terms of powers of . You have
where the last equality comes from the fact that . So
Now, setting , you get
Clearly, , so you're left with the quartic equation
Applying the quadratic formula gives a solution of
Since , we should expect to be smaller, which means we take the positive root because , and adding a positive number would make this larger. So,
but we also expect this number to be positive, so we ignore the negative root and end up with
So the limit is
Now, there's no reason to expect this to have a simpler form, so we can stop here. (Perhaps this answer is overkill, but if you didn't know this stuff, it doesn't hurt to learn it.)