(a)Yes, it is true that if lim f(x) = ∞, then lim 1/f(x) = 0.
(b) It's not always true that if lim f(x) = 0, then lim 1/f(x) = ∞.
a. As f(x) approaches infinity, its reciprocal 1/f(x) approaches 0. This is because as the magnitude of f(x) increases without bound, the magnitude of 1/f(x) decreases towards 0.
Let ε > 0 be given. Since lim f(x) = ∞, there exists M > 0 such that |f(x)| > M whenever x is sufficiently close to the limit point.
Choose M = 1/ε. Then, |1/f(x)| < ε whenever x is sufficiently close to the limit point.
Therefore, lim 1/f(x) = 0.
B Consider f(x) = 0 for all x. Then, lim f(x) = 0, but 1/f(x) is undefined.
However, if f(x) approaches 0 through non-zero values (i.e., f(x) ≠ 0 as x approaches the limit point), then lim 1/f(x) = ∞ or -∞, depending on the sign of f(x).
As f(x) approaches 0 through non-zero values, its reciprocal 1/f(x) increases or decreases without bound, depending on the sign of f(x).
Let M > 0 be given. Since lim f(x) = 0, there exists δ > 0 such that |f(x)| < 1/M whenever x is sufficiently close to the limit point.
Then, |1/f(x)| > M whenever x is sufficiently close to the limit point.
Therefore, lim 1/f(x) = ∞ or -∞, depending on the sign of f(x).
