1. Determine the following limits, if they exist. If you applyL’Hˆopital’s Rule, you must indicate the type of indeterminate formand each step you apply it.
(a) (3 points) lim x→0+ √ x /sin(x)
(b) (3 points) limx→∞ x^ 2 /e ^x
2. Let f(x) = x/ x ^2 − 9
(a) (1 point) Determine all intercepts of f. (x-intercepts andy-intercept. if they exist.)
(b) (1 point) Determine all vertical asymptotes of the graph off, if they exist.
(c) (1 point) Determine all horizontal asymptotes of the graphof f, if they exist.
(d) (3 points) Find all critical points of f.
(e) (3 points) Find where f is increasing and where f isdecreasing. State your answer using interval notation.
(f) (2 points) Find all local maximum and local minimum valuesof f. State each value, what type of extrema it is, and where itoccurs.
(g) (3 points) Find where f is concave up and where f is concavedown. State your answer using interval notation.
(h) (2 points) Find any inflection points of f. State youranswer as an ordered pair.
(i) (3 points) Use the information determined in parts (a)-(h)to sketch a graph of the function f(x) = x/ x^ 2 − 9 by hand. Nocredit will be given for submitting a graph created using agraphing program.
