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Worksheet for Week 4: Limits and Derivatives
This worksheet reviews limits and the definition of the derivative with graphs and computations.
- Answer the following questions using the graph y = f (x) below. The function f (x) has domain all numbers except 7 as seen from the graph.
(a) lim xโ 4
f (x) =
(b) lim xโ 7 +^
f (x) =
(c) f โฒ(0) =
(d) (^) xlimโโ 3 f (x) =
(e) lim xโ 0
f (x) x
(f) lim hโ 0
f (3 + h) โ 5 h
(g) f โฒ(5) =
(h) lim hโ 0 +
f (โ8 + h) โ f (โ8) h
(i) lim hโ 0
f (โ8 + h) h
(j) lim hโ 0
f (โ6 + h) โ f (โ6) h
(k) lim hโ 0 +
f (โ3 + h) + 5 h
(l) List all the intervals where the derivative f โฒ(x) is negative.
(m) List all the intervals where the derivative f โฒ(x) is decreasing.
(n) A critical value for f (x) is any x in the domain of f (x) where f โฒ(x) = 0 or f โฒ(x) is undefined. List all critical values of f (x).
- Evaluate the following limits and then match the functions with their graphs shown below using your limit results. Some will require you to compute left and right hand limits. (a) lim xโ 5
x โ 5
(b) lim xโ 5
โx (x โ 5)^2
(c) lim xโ 5
โx^2 โ 2 x + 35 x^2 โ 4 x โ 5
(d) lim xโ 5
x โ
3 x + 10 x โ 5
- Find, if any, the horizontal asymptotes of the following functions and use that information to match them with their graphs on the next page. Each question should have two limit computations with x โ โ and x โ โโ.
(a) f (x) =
(x + 1)^4 x^4 + 3x^2 + 7x + 10
(b) f (x) =
x + 3 x^2 + 8x + 26
(c) f (x) =
x^3 + 4x + 9 x^2 + 4
(d) f (x) = โ 7 x^4 + x^3 โ 12 x + 20
(e) f (x) =
8 x^2 + 4 x + 2
(f) f (x) = 3ex
(g) f (x) = 7 โ eโx
