Hello everyone!
So, this is the question, and I am not sure I am able to find any reason why this statement is wrong, as the exercise requires.
Exercise 2.3.53
Show by example that the following statement is wrong: The number L is the limit of
f(x) as x approaches x_{0} if f(x) gets closer to L as x approaches x_{0}.
Explain why the function in your example does not have the given value of L as a
limit as x -> x_{0}.
I have just started my university calculus course and am feeling pretty lost most of the time. Some words of advice would be nice if anyone has any hehe.
Thank you everyone!
Are you sure about that? If I was to approach 2, I'm sure my f(x) would approach 1.
I think the problem with the statement as given is that you need to approach L FROM BOTH SIDES as you make x approach x_0 FROM BOTH SIDES in order to have L be the limit of f(x) as x approaches x_0.
The idea is that you need to not only approach it, but it is also required that you get arbitrarily close to the limit. In the example stated above, in a manner of speaking, you are "getting closer to" 0 from both sides, but you never get closer than 1 unit away. So the limit is not 0.