Calculus: concept of a limit

Tutoring calculus, the idea of a limit is important. The tutor mentions some thoughts.

A limit, in calculus, is important, because it opens the door to certain functions, procedures, and assumptions. Numerous mechanisms are available to prove or disprove the existence of a limit depending on the situation.

The concept of a limit is important to understand, since it can guide one towards realizing whether one can be expected or not.

Let’s imagine that the limit of f(x) = b as x approaches a. The meaning is that f(x) will get closer to b as x gets closer to a. Moreover, we can reach a definite value for f(x) very close to b by setting x accordingly. People sometimes refer to this as “arbitrarily close.”

An example is that the limit of f(x) = x^2, as x approaches 3, is 9. What if we want x^2 to equal 8.99? We just set x to (8.99)^0.5. What about if we want x^2 to equal 8.9999? We can set x to (8.9999)^0.5. From the other side, we can reach 9.001 by setting x to (9.001)^0.5. The idea is that, as x closes in on 3 from either direction, x^2 converges towards 9. This might seem trivial, but it’s so easily illustrated because the limit of x^2 as x approaches 3 is 9.

If one tries to prove the the limit of 1/(x-5) as x approaches 5 exists, it can’t be done: as x gets closer to 5, 1/(x-5) grows in magnitude (negative if x<5, positive if x>5). Therefore, it doesn’t converge, but rather runs away. The limit of 1/(x-5) doesn’t exist as x approaches 5. However, the limit of 1/(x-5) does exist as x approaches 0: it’s -1/5.

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus, third edition. Toronto: D. C. Heath and Company, 1989.