Limits and function values

Suppose we wish to compute

where . Since limits only care about what happens as we approach , why do we then compute the limit by simply plugging in in ?

What if we were to discover that but going towards negative infinity and going towards , or something random like that? How can we be so sure of this not happening (without actually checking values close to )?

What's the justification for equating with ?

Re: Limits and function values

Quote:

Originally Posted by

**MathCrusader** What's the justification for equating

with

?

It is that f(x) is continuous at x= 1; therefore, its limit equals its value. The fact that f(x) is continuous is implied by the fact that f(x) is a composition of continuous functions. In particular, it is known that 1 / (x + 3) is continuous everywhere except when the denominator turns into 0.

Re: Limits and function values

If i remember right, if the fonction is continuous in f(a) then (lim x=> a) = f(a)

Re: Limits and function values

Quote:

Originally Posted by

**emakarov** It is that f(x) is continuous at x= 1; therefore, its limit equals its value. The fact that f(x) is continuous is implied by the fact that f(x) is a composition of continuous functions. In particular, it is known that 1 / (x + 3) is continuous everywhere except when the denominator turns into 0.

Ah, alright I understand! But suppose we didn't know that f(x) was continuous?

Re: Limits and function values

well you cant not know... you have to check it to evaluate the limit. You probably do it without even knowing.