Limits and Continuity
LIMITS AND CONTINUITY
1. The concept of limit x2 − 4 . Examine the behavior of f (x) as x approaches 2. Example 1.1. Let f (x) = x−2 Solution. Let us compute some values of f (x) for x close to 2, as in the tables below.
We see from the first table that f (x) is getting closer and closer to 4 as x approaches 2 from the left side. We express this by saying that “the limit of f (x) as x approaches 2 from left is 4”, and write x→2− lim f (x) = 4.
Similarly, by looking at the second table, we say that “the limit of f (x) as x approaches 2 from right is 4”, and write x→2+ x→2 x→2
lim f (x) = 4.
We call lim f (x) and lim f (x) one-sided limits. Since the two one-sided limits of f (x) are − + the same, we can say that “the limit of f (x) as x approaches 2 is 4”, and write x→2 lim f (x) = 4.
Note that since x2 − 4 = (x − 2)(x + 2), we can write x→2 lim f (x) = lim
x2 − 4 x→2 x − 2 (x − 2)(x + 2) = lim x→2 x−2 = lim (x + 2) = 4, x→2 where we can cancel the factors of (x − 2) since in the limit as x → 2, x is close to 2, but x = 2, so that x − 2 = 0. Below, find the graph of f (x), from which it is also clear that limx→2 f (x) = 4.
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LIMITS AND CONTINUITY
Example 1.2. Let g(x) =
x2 − 5 . Examine the behavior of g(x) as x approaches 2. x−2
Solution. Based on the graph and tables of approximate function values shown below,
observe that as x gets closer and closer to 2 from the left, g(x) increases without bound and as x gets closer and closer to 2 from the left, g(x) decreases without bound. We express this situation by saying that the limit of g(x) as x approaches 2 from the left is ∞, and g(x) as x approaches 2 from the right is −∞ and write x→2− lim g(x) = ∞,
x→2+
lim g(x) = −∞.
Since there is no common value for the one-sided limits of g(x), we say that the limit of g(x) as x approaches 2 does not exists and write x→2 lim g(x) does not exits. x→1 x→1 x→1 x→−1
Example 1.3. Use the graph below to
1. The concept of limit x2 − 4 . Examine the behavior of f (x) as x approaches 2. Example 1.1. Let f (x) = x−2 Solution. Let us compute some values of f (x) for x close to 2, as in the tables below.
We see from the first table that f (x) is getting closer and closer to 4 as x approaches 2 from the left side. We express this by saying that “the limit of f (x) as x approaches 2 from left is 4”, and write x→2− lim f (x) = 4.
Similarly, by looking at the second table, we say that “the limit of f (x) as x approaches 2 from right is 4”, and write x→2+ x→2 x→2
lim f (x) = 4.
We call lim f (x) and lim f (x) one-sided limits. Since the two one-sided limits of f (x) are − + the same, we can say that “the limit of f (x) as x approaches 2 is 4”, and write x→2 lim f (x) = 4.
Note that since x2 − 4 = (x − 2)(x + 2), we can write x→2 lim f (x) = lim
x2 − 4 x→2 x − 2 (x − 2)(x + 2) = lim x→2 x−2 = lim (x + 2) = 4, x→2 where we can cancel the factors of (x − 2) since in the limit as x → 2, x is close to 2, but x = 2, so that x − 2 = 0. Below, find the graph of f (x), from which it is also clear that limx→2 f (x) = 4.
1
2
LIMITS AND CONTINUITY
Example 1.2. Let g(x) =
x2 − 5 . Examine the behavior of g(x) as x approaches 2. x−2
Solution. Based on the graph and tables of approximate function values shown below,
observe that as x gets closer and closer to 2 from the left, g(x) increases without bound and as x gets closer and closer to 2 from the left, g(x) decreases without bound. We express this situation by saying that the limit of g(x) as x approaches 2 from the left is ∞, and g(x) as x approaches 2 from the right is −∞ and write x→2− lim g(x) = ∞,
x→2+
lim g(x) = −∞.
Since there is no common value for the one-sided limits of g(x), we say that the limit of g(x) as x approaches 2 does not exists and write x→2 lim g(x) does not exits. x→1 x→1 x→1 x→−1
Example 1.3. Use the graph below to