Limits and Graphical Behavior
Limits can tell a lot about a function’s behavior at some particular point. Then how should one use it? In cases where it is known that the function is discontinuous, it becomes hard to analyze the graph of the function solely using derivatives. In such cases, limits can be used to find the values of the function at the points of discontinuities. For example, let’s say we have a function f(x),
It is obvious from the function definition that this function is discontinuous, so derivatives are not useful at such places. Graph shapes for individual pieces are already known. Let’s calculate the value of the function at discontinuity x = 1.
Left-hand Limit at x =1,
Right-hand Limit at x =1,
⇒
⇒ 5
So, the values of both limits are different. Now we can plot the function on a graph.
Estimating Limits from Graphs
The concept of limits has been around for thousands of years. Earlier mathematicians in ancient civilizations used limits to approximate the area of a circle. But the formal concept was not around till the 19th century. This concept is essential to calculus and serves as a building block for analyzing derivatives, continuity, and differentiability. Intuitively, limits give us an idea about the values function approaches at a particular value of x. Using this idea, limits can also be estimated to a certain extent just by looking at the graph. Let’s look at these ideas in detail.
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