LCKurtz said:Did you miss the part about "using the limit laws" near the end? For example, the limit of a quotient is the quotient of the limits if the limit of the denominator isn't zero, etc...
lurflurf said:Yes, in the initial form the function is undefined when x=0. The new form is a continuous function so the value when x=0 is the same as the limit, which is the same as the original limit. It is also of interest to notice that the limit is a Newton quotient so its value is recognized as 3sqrt'(1).
Actually, the function they give us is NOT continuous because it is not defined at x= 0. Of course, IF the limit exists and we redefine the function to have that value at x= 0 then it is continuous at x= 0. The "limit theorem" used here is "if the sequence [itex]\lim_{n\to\infty}f(x_n)= L[/itex] for every sequence [itex]x_n[/itex] that converges to [itex]x_0[/itex], then [itex]\lim_{x\to x_0} f(x)[/itex] exists and is equal to [itex]L[/itex]".Artusartos said:Thanks...
This is what I understood, but I'm not sure if it is correct...
Since the function they are giving us is continuous, we know that for any sequence x_n in the domain that converges to 0, [tex]f(x_n) \rightarrow f(0)[/tex]. Is that the proof?
A limit in mathematics is a fundamental concept that describes the behavior of a function as the input approaches a specific value. It is used to determine the value that a function approaches as the input gets closer and closer to a certain value.
The existence of a limit is important because it allows us to understand the behavior of a function in a more precise and accurate way. It helps us to determine the exact value that a function approaches, rather than just estimating it based on its graph or table of values.
The existence of a limit is determined by evaluating the function at values that get closer and closer to the specific value in question. If the function approaches a single value as the input approaches the specific value, then the limit exists. If the function approaches different values from the left and right sides, then the limit does not exist.
Limits are used in many areas of science and mathematics, such as physics, engineering, and economics. They can be used to model and predict the behavior of systems, to optimize functions, and to analyze the growth and decay of populations.
Yes, a function can have a limit at a point where it is not defined. This is because the existence of a limit is dependent on the behavior of the function as the input approaches a specific value, not necessarily the value of the function at that point. However, if the function is undefined at the specific value, then the limit may not exist.