Suppose that limx→af(x)=0, limx→ag(x)=0, limx→ah(x)=1, limx→ap(x)=∞, and limx→aq(x)=∞. Evaluate each limits. a) limx→a[h(x)]^p(x) b) limx→a[p(x)]^f(x) c)limx→aq(x)sqrt(p(x))

Answer 1

We can be sure only that:

(c) lim q(x)√(p(x)) = ∞

x→a

For (a) and (b) limits can be any positive numbers.

Explanation. Famous calculus facts:

lim (1 + 1/x)^x = e, lim (1 + 1/x)^kx = e^k

x→∞ x→∞

lim exp(kx)^(1/x) = e^k (take a logarithm to verify this)

(a) Let h(x) = 1 + (x-a)² , p(x) = k/(x-a)²

Replace x with y = 1/(x-a)²

lim h(x)^p(x) = lim (1 + 1/y)^ky = e^k

x→a x→∞

(b) Let p(x) = exp(k/(x-a)²), f(x) = (x-a)²

Replace x with y = 1/(x-a)²

lim p(x)^f(x) = lim exp(ky)^(1/y) = e^k

x→a x→∞