need to find:
lim as x -> 0
of
4(e^2x - 1) / (e^x -1)
Try splitting the limit for the numerator and denominator
lim lim
x->0 4(e^2x-1) (4)x->0 (e^2x-1)
______________ = ________________
lim lim
x->0 e^X-1 x->0 e^x-1
Next solve for lim x->0 and simplify
To solve this limit, start by splitting the limit for the numerator and denominator separately.
lim as x -> 0 of 4(e^2x - 1) / (e^x - 1)
Splitting the numerator:
= lim as x -> 0 of 4(e^2x - 1)
And splitting the denominator:
= lim as x -> 0 of (e^x - 1)
Next, solve for each limit separately.
For the numerator limit:
lim as x -> 0 of 4(e^2x - 1)
= 4 * lim as x -> 0 of (e^2x - 1)
Now, solve for the limit of (e^2x - 1).
lim as x -> 0 of (e^2x - 1)
= (e^2*0 - 1)
= (e^0 - 1)
= (1 - 1)
= 0
So, the numerator limit becomes:
4 * lim as x -> 0 of (e^2x - 1)
= 4 * 0
= 0
Now, let's solve the limit for the denominator.
lim as x -> 0 of (e^x - 1)
= (e^0 - 1)
= (1 - 1)
= 0
Now, we can rewrite the original limit using the results from the numerator and denominator limits.
= (numerator) / (denominator)
= 0 / 0
Since this is an indeterminate form (0/0), we need to apply a different approach to evaluate this limit.
One way to find the limit in this case is to use L'Hôpital's rule, which states that if we have a limit in the form of 0/0 or ∞/∞, we can differentiate the numerator and denominator separately until we no longer get this indeterminate form.
Differentiating the numerator:
d/dx (4(e^2x - 1)) = 8e^2x
Differentiating the denominator:
d/dx (e^x - 1) = e^x
Now, let's find the limit again using the derivatives.
lim as x -> 0 of (8e^2x) / (e^x)
Plugging in x = 0:
= (8e^0) / (e^0)
= 8/1
= 8
So, the limit as x approaches 0 of 4(e^2x - 1) / (e^x - 1) is equal to 8.