Lesson 1.2: The Limit

From Algebra II, you will remember the informal definition of a limit: if a function f(x) approaches a value L while its input x approaches a value a, we write lim(x->a)f(x)=L and say “the limit of f of x as x approaches a equals L”. These limits can be finite or infinite, and the function may or may not actually reach the limit.

For example, consider f(x)=x^2. Then, lim(x->2)f(x)=f(2)=2^2=4. This may be informally demonstrated by examining the graph of y=f(x) and noting that f(2)=4 and f(x) is near 4 whenever x is near 2.

 Graph of y=f(x)=x^2
The graph of y=f(x)=x^2

Things are not always so simple, however. Take g(x)=sin(x)/x around x=0. Because division by zero is not allowed, g(x) is undefined at x=0. But if we graph y=g(x), it appears that lim(x->0)g(x)=1 (this is correct, which we will prove later in Chapter 2). Again using our informal definition of a limit, which we will soon make rigorous, it is clear that functions can have a limit where they are undefined.

 Graph of y=g(x)=sin(x)/x
The graph of y=g(x)=sin(x)/x

Now let’s examine h(x)=1/x^2. Again, h(x) is undefined at x=0. Looking at the graph of y=h(x), it seems that lim(x->0)h(x)=infinity. The limit doesn’t exist, but we often speak of a function having an infinite limit; that is, it gets larger and larger (or negatively so) without bound.

 Graph of y=h(x)=1/x^2
The graph of y=h(x)=1/x^2

The final type of limit we need to consider is the one-sided limit. Consider k(x)=1/x. Once again, k(x)is undefined at x=0, but if we look at the graph of y=k(x), it would be incorrect to say that lim(x->0)k(x)=infinity because this is only true if we approach x=0 from the positive x-axis. If we take x=1, x=1/2, x=1/4, x=1/8, etc., we can get as large a value of k(x) as we like; that is, k(x) approaches ∞. But if we take x=-1, x=-1/2, x=-1/4, x=-1/8, etc., k(x) gets infinitely small; that is, it approaches negative infinity. Since the value of the limit is different depending on which side it is approached from, we write lim(x->0+)k(x)=infinity and lim(x->0-)k(x)=-infinity. In general, we say that lim(x->a)b(x) exists if and only if lim(x->a+)b(x) = lim(x->a-)b(x) (and both, of course, exist).

 Graph of y=k(x)=1/x
The graph of y=k(x)=1/x

Occasionally, we find a function m(x) where the limit at a certain value (let’s call it a) of x exists, but m(a) is not equal to lim(x->a)m(x). This is fairly unusual; it most frequently occurs in piecewise-defined functions such as m(x) = x for x=2 and x^2 for all other values. In this case, m(2)=2, but as can be seen from the graph of y=m(x), lim(x->2)m(x)=2^2=4 which is not equal to 2.

Graph of y=m(x) = x for x=2 and x^2 for all other values
The graph of y=m(x) = x for x=2 and x^2 for all other values

After a brief diversion to the discussions of continuity and evaluating limits, we will arrive at a formal definition of a limit. While this intuitive concept of a limit is generally sufficient, the study of mathematics is concerned with rigor. Therefore, we will soon discuss the foundation of calculus: Augustin-Louis Cauchy’s delta-epsilon definition of the limit.

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