Final answer:
The limit of arctan(x) as x approaches 0 from the left is 0 because the arctan function is continuous at 0 and has an arctan value of 0 at that point.
Step-by-step explanation:
To evaluate limx → 0 arctan(x) from the left, we want to consider values of x that are approaching 0 from the negative side. The arctan function, also known as the inverse tangent function, is continuous around 0 and has a value of arctan(0) = 0. Therefore, as x approaches 0 from the left (meaning x is negative but getting closer to 0), the value of arctan(x) will approach 0.
Arctan is an odd function, meaning arctan(-x) = -arctan(x). Since the values of x we are considering are negative but getting increasingly closer to 0, the values of arctan(x) are also getting closer to 0 from the negative side. Thus, limx → 0⁻ arctan(x) = 0, where the superscript ‘⁻’ indicates the limit is being taken from the left.