Explain why the function is discontinuous at the given number a. (Select all that apply.)

Question

f(x) = 1/x+2, a = -2

These are the options for an answer...

A. f(−2) and lim x→−2 f(x) exist, but are not equal.

B. lim x→−2  f(x) does not exist.

C. lim x→−2⁺  f(x) and lim x→−2⁻  f(x) exist, but are not equal.

D. f(−2) is undefined.

E. none of the above

Answer 1

D) is correct. when the denominator = 0, the fraction is undefined

B) and C) are correct. The limit of f(x) as x approaches -2 from the left and right have different limits. One approaches negative infinity, the other side approaches positive infinity, which are very unequal. It's discontinuous at x=-2. since the limits are different there is no limit for when x approaches x=-2

This all assumes the problem was meant to say f(x) = 1/(x+2). But 1/x+2 does really mean (1/x)+2 if you follow the PEMDAS order of operations, D before A, division before addition. Then E) is the correct answer.

A) is the worst answer. as f(-2) DNE, it's undefined. anything divided by zero is undefined. Plus the right and left limits are not equal, so the limit as x approaches -1 DNE

The instructor who devised this question might hate his students. Or as one instructor said in another context, he made the questions hard to you wouldn't think math is easy.