3. (a) (i) -oo (ii) oo (iii) 0 (iv) A (b) lim,_0 fix) = B

5. (a) y = x — 1 (x = — 1 is a vertical asymptote), (b) y = Ix — 3 (c) y = 3* + 5 (x = 1 is a vertical asymptote), (d) y = 5x ix = 1 is a vertical asymptote).

1. (b) and (d) are continuous; the others are not necessarily so. (As for (c), consider Problem 6.)

3. / is discontinuous at x = 0. g is continuous at x = 2. The graphs of / and g are shown in Figs. 19 and 20.

5. a = 5, so that lim^i- f(x) = 4 = lim^i- f(x) = a — 1.

7. No. Let /(*) = g(x) = 1 for x < a, let /(*) = -1 and g(;c) = 3 for x > a. Then / and g are both discontinuous at x — a, but fix) + g(x) = 2 for all x, and therefore / + g is continuous for all x. (Draw a figure!) Let h(x) = —f(x) for all x. Then h is also discontinuous at x — a, whereas f(x)h(x) = — 1 for all x, and so / • h is continuous for all x.

3. If x > 0, then f'{x) = \x~2/3 oo as x 0+. Also f'(x) ^ -oo as x 0~. Hence, the graph has a cusp at x = 0. See Fig. 22.

1. (a) an =--► —- as n -> oo (b) 6n =------^ -

as /I oo (c) 3 - (-1/2) + 4 • (1/3) = -1/6 (d) (-1/2) • (1/3) = -1/6 (e) (-l/2)/(l/3) = -3/2 (f) V(1/3) — (—1/2) = V5?6 = ^730/6

Was this article helpful?

## Post a comment