Question

lim ((1/2+x)-(1/2))/x
x->0
find the limit

Answer 1

simplify and the answer will just opo out because there will be no more x's

Answer 2

ok how do you simplify that

Answer 3

\[\lim_{x \rightarrow 0}\frac{ \left( \frac{ 1 }{ 2 }+x \right) -\frac{ 1 }{ 2 }}{ x }\]right?

simplfy the numerator

Answer 4

how do i simplify the numerator?

Answer 5

Is his expression correct? is it \(\frac 12+x\) or \(\frac 1 {2+x}\)

Answer 6

\[\left( \frac{ 1 }{ 2 } + x\right) - \frac{ 1 }{ 2 }= \frac{ 1 }{ 2 } + x- \frac{ 1 }{ 2 }=x\]

Answer 7

so i'd end up with x/x, plug in 0 and get 0/0=0, which means the limit does not exist?

Answer 8

no... what's x/x?

Answer 9

from what you told me about how to simplify the numerator, I'd end up with x in the numerator as well as the denominator. at the point after simplfying, i'd have x/x according to your method of simplifying

Answer 10

\[
\frac x x = \begin{cases}
1 &x\neq 0\\
\text{undef} & x=0
\end{cases}
\]

Answer 11

exactly... now what is x/x? don't plug in 0, just simplify.

Answer 12

wio do you know what a limit is?

Answer 13

i have to plug 0 into the simplified answer, the question is asking for the limit as the graph approaches 0

Answer 14

yes, but because you never get to zero, you can cancel the x's to get x/x = 1. then take the limit which is just gonna be 1 because x is gone!

Answer 15

ok, i think i get it. sorry my calc teacher sucks and really doesnt bother to explain anything

Answer 16

that's the whole thing with limits... you can get rid of division by zero (if it's possible) by manipulating the expression.

Answer 17

and you only approach the value in the limit statement... you don't actually get there

Answer 18

@theblondeone5 The limit is the value such that:
- if you give me the maximum error between said value and f(x)
- I can give you you a domain of inputs in which all values of f(x) are within that error

Answer 19

It's an extremely abstract idea with a lot of pedantry that is often overlooked.

Answer 20

The question is: How do you draw this line without lifting your pen? Is it possible?
If it is possible, then what is the value of f(x) when you were at zero?

That is another way of thinking about limits.