can someone check my work on this relaively easy limit problem

5. Find all the local and absolute extrema (maxima and minima) for

f

(x) = x^3 +1 for x<1

1/x + 1 for x greater than or equal to 1

also calculate the limits as x goes to + and - infinity

i got local max when x=1

i dont think there are any absolute extrema

and i think the lim as x goes to +infinity is 0

and as x goes to -ininity is positive infinity...how do i show thats the case?

am i right on all counts?

unfortunately ill be on here for the next 2-3 hours or so, so if you could help me with any of the questions ive yet asked id be extremely grateful (with my last and worst one still to come:()