If you're shocked by that limit, then it's clear that you don't understand the definition of limits.appplejack said:Homework Statement
Evaluate the limits
lim x->2 f(x) if f(x) = 3, x an integer, or 1, otherwise.
The Attempt at a Solution
I just 3 was the answer because I thought 2 is an integer and if x is an integer then the answer is 3. I was rather shocked to find that the answer is 1. I understand the definition of limits but I think I'm obviously mistaken about something. Thanks.
appplejack said:lim x->c f(x)=L
as 'x' approaches c 'a number' from both sides (-,+) and F(x)= approaches L from both side (-,+).
A limit is a mathematical concept that describes the behavior of a function as the input approaches a certain value or point. It represents the value that the function is approaching, or the value that the function gets closer and closer to, as the input gets closer and closer to a specific value.
To evaluate a limit, you can use various techniques such as direct substitution, factoring, or algebraic manipulation. You can also use the properties of limits, such as the sum, difference, product, and quotient rules, to evaluate more complex limits.
A one-sided limit, also known as a unilateral limit, is a limit where the input approaches the specified point from only one direction. It is written as either f(x) → L as x approaches a+, or f(x) → L as x approaches a-. This is different from a two-sided limit, where the input approaches the specified point from both the left and the right.
A limit at infinity is a special type of limit where the input approaches positive or negative infinity. It is written as either f(x) → L as x approaches positive infinity, or f(x) → L as x approaches negative infinity. This type of limit is used to describe the long-term behavior of a function as the input gets larger or smaller without bound.
Limits are important in mathematics because they help us understand the behavior of a function and its graph. They are used to define continuity, derivatives, and integrals, which are fundamental concepts in calculus. Limits are also used in many other areas of mathematics, such as in the study of sequences and series, and in the analysis of functions and their properties.