Limits describe the behavior of a function as it approaches a specific point, often used for finding the rate of change of a function.
A limit describes the value that a function approaches as the input (usually x) gets closer and closer to a particular value.
Consider the function $f(x) = \frac{x^2 - 1}{x - 1}$. As x approaches 1, this function gets closer and closer to 2, even though it's undefined at x = 1. We write this as:
$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$$
The limit of a function as it approaches a point is not the same as the function at that point. We do not care about the value of the function at that point, only about its behavior as it approaches that point.
Students learn to estimate limits using tables of values and graphs. This approach helps build intuition about function behavior near critical points.
To estimate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$, we can create a table:
From this, we can estimate that the limit as x approaches 2 is 4.
When estimating limits from graphs, look at the y-values the function approaches as x gets very close to the point of interest from both sides.
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