Math 1111 Calculus (I)
Homework 1
1. For the function h whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.
(a) limx→−3−h(x) (b) limx→−3+h(x) (c) limx→−3h(x) (d) h(−3) (e) limx→0−h(x) (f ) limx→0+h(x) (g) limx→0h(x) (h) h(0) (i) limx→2h(x) (j) h(2)
(k) limx→5+h(x) (l) limx→5−h(x)
2. For the function f whose graph is shown, state the following
(a) limx→−7f (x) (b) limx→−3f (x) (c) limx→0f (x) (d) limx→6−f (x) (e) limx→6+f (x) (f) The equations of the vertical asymptotes.
3. Sketch the graph of the function and use it to determine the value of each limit, if it exists. If it does not exist, explain why.
(a) f (x) = |x|; limx→0f (x), (b) f (x) = x
|x|; limx→0f (x),
(c) f (x) =
1 + x if x < −1 x2 if − 1 ≤ x < 1 2 − x if x ≥ 1
; limx→−1f (x) and limx→1f (x),
(d) f (x) = 2 x rational
−2 x irrationl ; limx→2f (x)
4. The symbol [x] denotes the largest integer which is less than or equal to x. For example, [2.1] = 2, [3] = 3, [−0.5] = −1, [−2] = −2. Draw the graph of the following functions and determine the value of each limit
(a) f (x) = [x]; limx→1f (x).
(b) f (x) = x − [x]; limx→1f (x).
5. Let f (x) = x2. Determine the values of the limits (a) limx→3f (x).
(b) limh→0f (3 + h).
6. Determine the infinite limit (a) lim
x→1
2 − x (x − 1)2 (b) lim
x→2−
x2− 2x x2− 4x + 4
