Question

For each of the following functions f(x) below : Find its limits limx→±[infinity]f(x) as x approachs ±[infinity] . Choose the values of x where f(x) is differentiable, i.e. f′(x) exists Choose the values of x where f(x) is also strictly increasing, i.e. f′(x)>0 . For f(x)=max(0,x) : (If the limit diverges to infty, enter inf for [infinity], and -inf for −[infinity] ) limx→−[infinity]f(x)=

Answer 1

Answer:

See below

Step-by-step explanation:

$f(x)=\max\{0,x\}$ then f(x)=0 if x<0 and f(x)=x if x≥0.

When x→-∞, f is the constant function zero, therefore $\lim_{n\rightarrow -\infty}f(x)=0$. When x→∞, f(x)=x and x grows indefinitely. Thus $\lim_{n\rightarrow \infty}f(x)=\infty$

f is differentiable if x≠0. If x>0, f'(x)=1 (the derivative of f(x)=x) and if x<0, f'(0)=0 (the derivative of the constant zero). In x=0, the right-hand derivative is 1, but the left-hand derivative is 0, hence f'(0) does not exist,

f'(x)>0 for all x>0. Therefore f(x) is strictly increasing on the inverval (0,∞).