Introduction to Limits

Limits—Notation and Definition Definition of the limit of f(x) as x approaches a: We write and say “the limit of f(x), as x approaches a, equals L” if x gets closer and closer to a (from both sides of a but not equal to a), the values of f(x) (i.e., the y-values) are getting closer to L. Notice that x  a is in the definition of limit. This means: when we are finding the limit of f(x) as x approaches a, we never need to consider x = a. If fact, f(x) does not need to be defined at x = a. It is how f(x) is defined near a that matters. The following are three cases: (a) y y = f(x) (b) y (c) y y = f(x) y = f(x) L L L x x x a a a Figure 1 limxa f(x) = L In all three cases above, limxa f(x) = L. Case (a): limxa f(x) = L because as x approaches a, the values of f(x) are getting closer to L and f(a) is L. Case (b): limxa f(x) = L because as x approaches a, the values of f(x) are getting closer to L despite f(a) is undefined. Case (c): limxa f(x) = L because as x approaches a, the values of f(x) are getting closer to L despite f(a) is defined to be a number other than L.

Limit Does Not Exist? When we talk about the limit of a function f(x) as x approaches a, we must look at what the function values are getting closer to as x approaches a from both left and right. y = f(x) 4 y x 3 2 Figure 2 limx4 f(x) = DNE If, as x approaches a from the left, the function values are getting closer to a number M, where as, as x approaches a from the right, the function values are getting closer to another number N, we say the limit does not exist, or simply DNE. For example, in Figure 2, as x approaches 4 from the left, the function values are getting closer to 2, where as, as x approaches 4 from the right, the function values are getting closer to 3, therefore limx4 f(x) does not exist because 2 and 3 are not the same number. Recall that some functions have vertical asymptotes (VA). If a function has a VA at x = a, at least one of the following must occur: 1. As x approaches a from the left, the values of f(x) are going to  (or –). 2. As x approaches a from the right, the values of f(x) are going to  (or –). For example, in Figure 3, we can see that, as x approaches 4 from the left, the function values are going to , where as, as x approaches 4 from the right, the function values are going to –, so limx4 f(x) DNE. In Figure 4, we can see that, as x approaches 4 from the left, the function values are going to , where as, as x approaches 4 from the right, the function values are going to 1, so limx4 f(x) DNE. y = f(x) y x Figure 3 limx4 f(x) = DNE 4 y = f(x) y x Figure 4 limx4 f(x) = DNE 4 1

Infinite Limits vs. Limits at Infinity What if as x approaches a from both the left and the right, the function values are going to ∞ (as shown in Figure 5)? Well, we say limxa f(x) = . Similarly, if as x approaches a from both the left and the right, the function values are going to –∞ (as shown in Figure 6), we say limxa f(x) = –. The above two examples are called infinite limits since the limit of the function is infinity. What if x approaches ∞ (or –∞) instead of approaching some real number a? Well, we call these limits at infinity. For example, in Figure 7, we can see that limx∞ f(x) = 1 and limx–∞ f(x) = . We can see that if the limx∞ f(x) (or limx–∞ f(x)) is a real number, a horizontal asymptote can be also drawn. Hence, in graphs, infinite limits associate with vertical asymptotes and limits at infinity associate with horizontal asymptotes. Let’s find the following limits: 1. limx–∞ f(x) = 0 2. limx–4 f(x) = DNE 3. limx–1 f(x) = 0 4. limx0 f(x) = –1 5. limx1 f(x) = DNE 6. limx2 f(x) = 0 7. limx3 f(x) = 2 8. limx5 f(x) =  9. limx6 f(x) = 2 10. limx8 f(x) = DNE 11. limx9 f(x) = –3 12. limx∞ f(x) =  y = f(x) y x Figure 5 limxa f(x) = ∞ a y = f(x) y x Figure 6 limxa f(x) = –∞ a y = f(x) y x Figure 7 1 3 2 1 –1 –2 –3 4 5 6 7 8 9 10 11 –4 –5 –6

1. If limxa– f(x) = limxa+ f(x) = L, then limxa f(x) = L. One-Sided Limits As you recall, when we talk about the limit of a function f(x) as x approaches a, we have to look at the values of x approaching a from both left and right. However, we can also look at the values of x approaching a from one side only—this is called one-sided limit. If we only look at the values of x approaching a from the left side only, it’s called the left-sided limit or left-hand limit, and similarly, if we only look at the values of x approaching a from the right side only, it’s called the right-sided limit or right-hand limit. Left-sided limit: lim f(x) = L Definition: As x gets closer and closer to a from the left, and remains less than a, the values of f(x) (i.e., the y-values) are getting closer and closer to L. Right-sided limit: lim f(x) = L Definition: As x gets closer and closer to a from the right and remains greater than a, the values of f(x) (i.e., the y-values) are getting closer and closer to R. x→a– x→a+ 2 3 1. 2 3 2. 1 Properties: 1. If limxa– f(x) = limxa+ f(x) = L, then limxa f(x) = L. 2. If limxa– f(x)  limxa+ f(x), then limxa f(x) DNE. x→2– lim f(x) = 3 lim f(x) = 3 x→2+ x→2– lim f(x) = 3 lim f(x) = 1 x→2+ lim f(x) = 3 x→2 lim f(x) = DNE x→2

Implication of Limits on Continuity Whether the limit of f(x) as x approaches a exists, together with, whether the function f(x) is defined at a, has an important implication to another concept in calculus called continuity. Definition of f(x) is continuous at a number a: We say f(x) is continuous at x = a if limxa f(x) = f(a). The above definition requires (all of) the following three conditions be satisfied: 1. f(a) is defined, i.e., f(a) must be some real number, 2. limxa f(x) must exist, i.e., limxa f(x) must be some real number, and 3. limxa f(x) = f(a), i.e., the limit of the function at a must equal to the function value at a. A note on the 2nd condition: If limxa f(x) = ∞ or = –∞, it’s considered to be DNE. If all three conditions are satisfied, then f(x) is continuous at a. If one (or more) of the three conditions fails, we say f(x) is discontinuous at a. For example, in Figure 8, f(x) is continuous at 0 since f(0) is defined because f(0) = –1, limx0 f(x) exists because it’s equal to –1, and lastly, limx0 f(x) = f(0) because they are both equal to –1. Of course, f(x) is continuous at many other numbers too, to name a few, –3, 4, 5.6 and . Therefore, instead of naming the numbers f(x) is continuous at, we name the numbers f(x) is discontinuous at. In Figure 8, we can see that f(x) is discontinuous at x = –4, –1, 1, 3, 5, and 8, and let’s discuss why f(x) is discontinuous at these numbers. 3 2 1 –1 –2 –3 4 5 6 7 8 9 10 11 –4 –5 –6 Figure 8 y = f(x) x The Three Properties –4: (1) Y (2) N (3) N –1: (1) N (2) Y (3) N 1: (1) N (2) N (3) N 3: (1) Y (2) Y (3) N 5: (1) N (2) N (3) N 8: (1) Y (2) N (3) N

Three Types of Discontinuities y x Figure 9 f(x) = 2 x2 – 3x + 2 x – 2 1 (2,1) If we have the graph of a function, we can tell it is discontinuous when there are holes, gaps and/or vertical asymptotes. Each of these features is a different type of discontinuity. Removable Discontinuity Recall the three conditions on page 5 for a function f continuous at a number a—i) f(a) must be defined, ii) limxa f(x) must exist, and iii) limxa f(x) = f(a). If any one of these three conditions fails, f is said to be discontinuous at a. However, if condition 2 passes (i.e., limxa f(x) exists) while the 1st or the 3rd condition fails, then we said f has a removable discontinuity at a. Figures 9 and 10 illustrate this. In Figure 9, although the limx2 f(x) exists since it’s 1, but f(2) is undefined (i.e., condition 1 fails), therefore f has a removable discontinuity at x = 2. In Figure 10, limx2 f(x) is 1 and f(2) is defined to be 2. However, since limx2 f(x)  f(2) (i.e., condition 3 fails), therefore f also has a removable discontinuity at x = 2. This type of discontinuity is called removable because the discontinuity can be removed to make f continuous at a. For example, to make f continuous at 2, i) we can define the function-value at 2 to be 1 as in Figure 9, or ii) we can redefine the function-value at 2 to be 1 as in Figure 10. y x Figure 10 f(x) = 2 x2 – 3x + 2 x – 2 1 (2,1) (x  2) (x = 2) (2,2) If f is discontinuous at a, yet there is no number we can assign or reassign to f(a) to make f continuous, then f is said to have a non-removable (or essential) discontinuity at a (see Figures 11 and 12). Our textbook distinguishes the two types of discontinuities by calling the one in Figure 11 a jump discontinuity (because it “jumps” from one value to another) and the one in Figure 12 an infinite discontinuity (because of the vertical asymptote). In layman’s terms: The Three Types of Discontinuities Feature on the Graph 1. Removable Discontinuity Hole 2. Jump Discontinuity Gap 3. Infinite Discontinuity Vertical Asymptote y x a Figure 11 Figure 12 y x a

Continuity—On All Real Numbers Definition of f(x) is continuous everywhere: We say a function f(x) is continuous on all real numbers (or everywhere) if f(x) has no points of discontinuity on the real number line. Some functions are continuous everywhere while others are not. Examples of functions that are continuous everywhere: 1. Trig. functions such as y = sin x and y = cos x 2. Polynomial functions 3. Rational functions where domain is all real numbers 4. Exponential functions 5. Absolute-value and odd-indexed root functions Examples of functions that are not continuous everywhere: 1. Trig. functions such as y = tan x and y = sec x 2. Square root and even-indexed root functions 3. Rational functions where domain is not all real numbers 4. Logarithmic functions 5. Integer (a.k.a. step) functions [Graphs to come]