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Math 201: 1.7/2.1 Calculator limits Day 6
(1) The first part of this worksheet asks you to use your calculator and write down the answers you get. (a) Use your calculator to “expand” the function p(x) = −(x − e)(x + e)x (I mean the number e). (b) Graph this function in the window for x ∈ [− 5 , 5] and y ∈ [− 10 , 10]. Sketch the graph. (c) Use the table on your calculator to find p(.5), p(1), p(1.5), p(2), p(2.5). (d) Use your graph to find the x and y value at the “peak” in your sketch.
(2) Now you will investigate limits numerically (i.e. with calculators).
Definition 1. Assume f is defined near the x-value c (but not necessarily at c). Then the limit of f (x) as x approaches c is equal to L if |f (x) − L| is arbitrarily close to zero when x gets close to c. We write lim x→c f (x) = L.
Practically this means that if we plug in x’s getting closer and closer to c in the function f (x) and observe what number f (x) is approaching, this number that f (x) is approaching is the limit. Let’s try it. (a) Enter the function r(x) = 12 x
2 x^2 (x−5) into the “y^ equals” menu of your calculator and graph it. Use the “Value” command in the “Math” menu to complete the following tables.
x r(x) 1
1
01
001
0001
x r(x) − 1
− 0. 1
− 0. 01
− 0. 001
− 0. 0001
(b) As x gets close to zero, what value does r(x) approach? This is lim x→ 0
r(x). Notice that you need both tables to answer this question. (c) Is r(0) defined? If so, what is r(0)? 1
(^2) (d) Let’s do the same thing for the function g(x) = 5x + x− 4 |x− 4 |.^ Use your calculator to complete the following tables. x g(x) 5
1
01
001
0001
x g(x) 3
9
99
999
9999
(e) As x gets close to 4, what value does g(x) approach? (f) What value does g(x) approach as x approaches 4 from numbers greater than 4? This is a one-sided limit, lim x→ 4 +^
g(x). Notice that you only need the left table for this question. (g) What value does g(x) approach as x approaches 4 from numbers less than 4? This is a one-sided limit, lim x→ 4 −^
g(x). Notice that you only need the right table for this question. (h) In order for the full fledged limit lim x→ 4
g(x) to exist, both of the one-sided limits must exist and be equal. Do the limits lim x→ 0
r(x) and lim x→ 4
g(x) exist?
(i) We can also calculate limits at infinity. Let k(x) = 6 sin
(^2) (x) x + 13. Use your calculator to complete the following tables. x k(x) 10
100
1000
10000
100000
x k(x) − 10
− 100
− 1000
− 10000
− 100000
(j) As x gets really big, what value does k(x) approach? This is the limit lim x→∞
k(x). (k) As x gets really close to negative infinity, what value does q(x) approach? This is the limit lim x→−∞
k(x).
Some things to notice and remember:
f (x) does not depend on what happens to f (x) at the x-value c, but only near the x-value c.
- In order for the limit lim x→c f (x) to exist both one-sided limits, lim x→c+^
f (x) and lim x→c−^
f (x) must exist and be equal.
