1. Determine the following limits, if they exist. If you apply L’Hˆopital’s Rule, you must indicate...

1. Determine the following limits, if they exist. If you apply L’Hˆopital’s Rule, you must indicate the type of indeterminate form and 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 and y-intercept. if they exist.)

(b) (1 point) Determine all vertical asymptotes of the graph of f, if they exist.

(d) (3 points) Find all critical points of f.

(e) (3 points) Find where f is increasing and where f is decreasing. State your answer using interval notation.

(f) (2 points) Find all local maximum and local minimum values of f. State each value, what type of extrema it is, and where it occurs.

(g) (3 points) Find where f is concave up and where f is concave down. State your answer using interval notation.

(h) (2 points) Find any inflection points of f. State your answer 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. No credit will be given for submitting a graph created using a graphing program.

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Expert Solution

milcah answered 1 year ago

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