Find the limit as x approaches 0: (e^x^2-1)/(2x^2)
a. 0
b. 1/2
c. infinity
d. does not exist
e. none of the above
To find the limit as x approaches 0 for the given expression (e^x^2 - 1)/(2x^2), we can use the concept of L'Hôpital's Rule.
L'Hôpital's Rule states that if we have a limit of the form 0/0 or ∞/∞, and we can take the derivative of both the numerator and denominator, then the limit can be found by taking the derivative of the numerator and denominator and then calculating the limit again.
Let's apply L'Hôpital's Rule to find the limit:
First, let's differentiate the numerator:
d/dx (e^x^2 - 1) = 2x * e^x^2.
Next, let's differentiate the denominator:
d/dx (2x^2) = 4x.
Now, we have a new limit:
lim(x→0) (2x * e^x^2)/(4x).
We can simplify this expression by canceling out the common factor of 2x from the numerator and denominator:
lim(x→0) (e^x^2)/(2).
Now, let's substitute x=0 into the simplified expression:
(e^(0^2))/(2) = e^0/2 = 1/2.
Therefore, the limit as x approaches 0 for the given expression (e^x^2 - 1)/(2x^2) is equal to 1/2.
Thus, the answer is (b) 1/2.