# Thread: [SOLVED] Epsilon Delta (limit definition)

1. ## [SOLVED] Epsilon Delta (limit definition)

I'm given an epsilon value of 0.5
The limit of my function evaluated as x-->0 is 1

my function is

f(x) = (e^x - 1) / (x)

Just a restatement of the variable
e = 0.5
L = 1
f(x) = (e^x - 1) / (x)
x = ?
d(delta) = ?
a = 0

I've plugged the values into the inequalities and i'm stuck on this part:

|f(x)-L| < e
i.e.
| [(e^x - 1) / (x)] - 1| < 0.5
So, I simplify and then I get to this point:
e^x - 1 - 1.5x < 0
I can't figure out x and because I can't figure out x I'm unable to figure out delta. I need some help solving for x. I'm aware that I will likely need to apply a natural logarithm but whenever I do I still am unable to solve for x.

2. Originally Posted by Noxide
I'm given an epsilon value of 0.5
The limit of my function evaluated as x-->0 is 1

my function is

f(x) = (e^x - 1) / (x)

Just a restatement of the variable
e = 0.5
L = 1
f(x) = (e^x - 1) / (x)
x = ?
d(delta) = ?
a = 0

I've plugged the values into the inequalities and i'm stuck on this part:

|f(x)-L| < e
i.e.
| [(e^x - 1) / (x)] - 1| < 0.5
So, I simplify and then I get to this point:
e^x - 1 - 1.5x < 0
I can't figure out x and because I can't figure out x I'm unable to figure out delta. I need some help solving for x. I'm aware that I will likely need to apply a natural logarithm but whenever I do I still am unable to solve for x.
Note that we want $\displaystyle |x - 0| = |x| < \delta$. Lets say we want $\displaystyle \delta$ to be at most 1. so $\displaystyle |x| < 1 \implies -1 < x < 1 \implies e^{-1} < e^x < e$.

Now, go back to our expression and apply this bound on $\displaystyle e^x$:

$\displaystyle \bigg| \frac {e^x - 1}x - 1 \bigg| < \bigg| \frac {e - 1}x - 1 \bigg| < 0.5$

no solve the last inequality so that $\displaystyle x$ is bounded between two constants. $\displaystyle \delta$ will be the larger one

3. Thanks again, I really appreciate your help.