Title:

Author:

Introduction to Calculus

Copyright:

T. Tsujioka

Advanced Seminar

Introduction to Calculus

Part 1: Differential Calculus

Chapter 1. Limits

In this Introduction to Calculus, we offer the subject intuitively rather than rigorously so that you will grasp the concept quickly. Since functions include only polynomial, rational, and irrational ones, excluding trigonometric, exponential, logarithmic functions, even students who have not studied those transcendental functions will be able to study the content with ease --- step by step.

1-1. Finding Limits of Functions, Continuity

When the value of a function *f*(*x*) approaches to a certain finite definite value *L* as *x* approaches to a certain point a,

it is said that *f*(*x*) has a limit *L*, and write as follows:

If there is no such definite values, the function becomes infinite or has a discontinuous point. The condition of **continuity** at

*x* = *a*, thus, is that the limit exists and is equal to the value of the function at *x* = *a* :

In Fig. 1-1, the graph of *f*(*x*) is continuous, so that the limit is equal to the value of *f*(*x*) at *x* = *a*. In Fig. 1-2, when *x* approaches *a* from both sides, the value of *f*(*x*) becomes infinite --- there is no value specified. In Fig. 1-3, when *x* approaches *a* from the left side, the limit exists and coincides with the value of *f*(*x*) at *a* = *f*(*a*). On the other hand, when *x* approaches *a* from the right side, the limit exists, but no value of *f*(*x*) at *a*.

__Illustrated Examples 1-1. __:

Illustrated 1. [Explanations and Answers]

Illustrated 2. [__Explanations and Answers__]