3 Curve Sketching 1: Limits and Asymptotes K ELVIN S OH L AST UPDATED F EBRUARY 14, 2015
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3.1 An intuitive introduction to limits. Consider the relation y = x 2 + 1. If x = 1, then y = 2; this is probably not very interesting to many of us. Now let us frame our discussion differently. As x gets closer and closer to 1 (for example, we can think of the sequence 0.9, 0.99, 0.999, . . .), y will get closer and closer to 2 (1.81, 1.98, 1.998, . . . in our example). In fact, y can get arbitrarily close to 2 by choosing x sufficiently close to 1. We will call this process of arbitrarily approaching a number the limit. In symbols, we write as x → 1, y → 2. This is read “as x tends to 1, y tends to 2. Alternatively, we also use the notation lim (x 2 + 1) = 2,
x→1
read as “the limit of x 2 + 1 as x tends to 1 is 2.” So why do we introduce this notion of the limit? After all, substituting x = 1 into our given relation to obtain y = 2 is so much easier. For one, the behavior of limits is more mathematically interesting than simply substituting numbers into formulas. It also allows us to discuss many more concepts (such as asymptotes later in the chapter). In fact, the justification of modern calculus stems from the limit. Unfortunately (or fortunately for some students), this will not be covered in too much depth at our level. For some analogies of the limit to better understand why we want to introduce such a concept, I recommend the first half of the article at http://betterexplained.com/articles/an-intuitive-introduction-to-limits/ The second half of the article also introduces the precise definition of the limit (in ²−δ notation) for those who are interested or for those who find the description of limits using imprecise words like “arbitrarily close” unsatisfying.
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On a more practical level, we may use limits instead of substituting values because some functions may be undefined at certain points. For example, consider the function y = f (x) = e
−
1 x2
,
x ∈ R, x 6= 0.
y
y =1 y = f (x)
x
0
Even though we cannot substitute x = 0 into our formula (since division by 0 is undefined), by observing the graph, we can say that y → 0 as x → 0. Equivalently, lim e x→0
−
1 x2
= 0.
3.1.1 Limits from different sides Sometimes different behaviors emerge depending on how we approach a number. For example, consider the function: if x < 1 2x f (x) =
2.3 6 − 3x
if x = 1 if x > 1
y 3 2.3 2
0
y = f (x)
x 1
2
This function sure behaves weirdly. When x = 1, y = 2.3 but the value of 2.3 sure did come out of the blue. Instead, there is a certain behavior on the left and right sides of x = 1. As x approaches 1 “from the left” (i.e. taking x closer and closer to 1 using only numbers smaller than 1), we see that y approaches 2. We denote this by y → 2 as x → 1−
lim f (x) = 2.
or
x→1−
As x approaches 1 “from the right” (i.e. taking x closer and closer to 1 using only numbers bigger than 1), we see that y approaches 3. We denote this by y → 3 as x → 1+
lim f (x) = 3.
or
x→1+
3.1.2 Limits and infinity The use of limits allows us to tackle the concept of infinity. We say that x → +∞ if x gets arbitrarily large (and positive). For example, considering the function f (x) = e
−
1 x2
considered earlier, y → 1 as x → +∞
or
lim e
−
1 x2
x→+∞
= 1.
Remark: We often use just ∞ to denote +∞. Meanwhile, we say x → −∞ if x gets arbitrarily negative. Be mindful when using the ∞ notation. We should use the “tends to” arrows or the “limit” notation when handling infinity. For our syllabus, we avoid writing expressions like x = ∞ as many rules of arithmetic fail when introducing infinity as a number. For now, try to utilize infinity as a concept rather than as a number. To compute limits, we can either use graphs, use our calculators to investigate behaviors or argue analytically. When infinity is involved, some computations get tricky; practice will be necessary to familiarize ourselves with some of the concepts. Take note that forms such as ∞ − ∞,
∞ , ∞
0 , 0
0·∞
are indeterminate. If we encounter such expressions in our study, we should always try to rearrange our expression. This will be discussed in exercises at the end of this subsection: 3
Example: limits at infinity. Sketch the graphs of y = ex , y = ln x and y = x1 . If applicable, write down the limits relating to the 3 functions when x → ∞, x → −∞, x → 0+ and x → 0− .
Exercise: Compute the following limits • lim x 2 − 3x + 2 x→∞
1 x→∞ x − 99999
• lim
• lim
x +1
x→∞ 2x − 3
• lim
x +1
x→∞ x 2 + 2x − 3
x 2 + 2x − 3 x→∞ x +1
• lim
• lim (x + 1)e−x x→∞
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3.2 Asymptotes. We are often interested in “infinity behaviors” in many situations. For example, after leaving a hot cup of coffee on the table, we may be interested in its eventual temperature after much time have passed. When designing an experiment, we may be interested the conditions that cause pressure to be really, really high. (Can you spot where infinity arises for both examples?) The previous examples encountered in this chapter shows that these behaviors can be visualized as asymptotes. If y → a as x → ∞ for some real number a, then the line y = a is a horizontal asymptote to the graph of y against x. Similarly, if y → b as x → −∞, then the line y = b is also a horizontal asymptote. On the other hand, if y → ∞ as x → c for some real number c, then the line x = c is a vertical asymptote to the graph of y against x. Vertical asymptotes also arise if the above limits are modified to contain x → c + , x → c − or y → −∞.
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