Part 3/5:
Now, let's extend our understanding of limits beyond simple sequences. For a function ( y = f(x) ), the limit of ( f(x) ) as ( x ) approaches a specific value ( a ) is defined based on the values that f(x) takes as x comes near ( a ) from both directions (the left and the right).
The limit exists when:
[
\lim_{x \to a} f(x) = f(a)
]
For example, if we denote the left-hand limit as ( f(a^-) ) and the right-hand limit as ( f(a^+) ), the limit at ( a ) exists when these two conditions are equal:
[
\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)
]