Definition: Limits can often appear complex, but in this chapter, we will demystify them and make the concept more approachable. We will start by exploring limits in various scenarios, beginning with infinite limits ( – ∞ and + ∞ ) and then delve into finite limits.
PART 1: Infinite Limits
Limits of trigonometric functions like sin(x) and cos(x) as they approach + ∞ or – ∞ do not exist; they are undefined. This means that these functions do not have a specific value as x gets infinitely large or infinitely small.
| lim sin(x) = undef |
| x-> -∞ |
| lim sin(x) = undef |
| x-> +∞ |
| lim cos(x) = undef |
| x-> -∞ |
| lim cos(x) = undef |
| x-> +∞ |
PART 2: Finite Limits
Now, let’s explore finite limits in more detail. When a function has a finite limit at a point a0, it indicates that the function approaches a specific value as it gets closer to that point. This uniqueness of the limit is a crucial property of mathematical functions.
For instance, consider the limit of the sine function:
| lim sin(x) = sin(a0) |
| x -> a0 |
The graph of sin(x) demonstrates this concept as it approaches a specific value as x approaches a0:
 |
If you want to understand this concept further, you can refer to the trigonometry circle for insights into the limits of trigonometric functions: Limits of Trigonometry in the UNIT CIRCLE