Finding the limit of a piecewise function at an integer point

[appplejack]

Homework Statement

Evaluate the limits
lim x->2 f(x) if f(x) = 3, x an integer, or 1, otherwise.

Homework Equations

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.

 

[SammyS]

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.

If you're shocked by that limit, then it's clear that you don't understand the definition of limits.

 

[vela]

What is your intuitive understanding of what a limit is, appplejack?

 

[appplejack]

lim x->c f(x)=L
as 'x' approaches c 'a number' from both sides (-,+) and F(x)= approaches L from both side (-,+).

 

[Dick]

lim x->c f(x)=L
as 'x' approaches c 'a number' from both sides (-,+) and F(x)= approaches L from both side (-,+).

That's about right. But the definition of a limit doesn't say anything about the value of f(c) does it? It just talks about the value of f(x) where x is 'close to c'. Not equal to c.