Use continuity to evaluate. as x approaches 2, lim arctan((2x^2-8)/(3x^2-6x))
To evaluate the limit as x approaches 2, we can use continuity and substitute the value into the function. Here's how we can do it step by step:
First, let's simplify the function inside the arctan function:
f(x) = arctan((2x^2 - 8) / (3x^2 - 6x))
Now, let's substitute x = 2 into the function:
f(2) = arctan((2(2)^2 - 8) / (3(2)^2 - 6(2)))
= arctan((2(4) - 8) / (3(4) - 12))
= arctan((8 - 8) / (12 - 12))
= arctan(0 / 0)
At this point, we have an indeterminate form of 0/0. This means we need to further simplify the expression to evaluate the limit. To do so, we can factor out the common term:
f(2) = arctan(0 / 0)
= arctan(0)
The arctan(0) evaluates to 0, so the limit as x approaches 2 of the given function is 0.
In summary, we used the concept of continuity to substitute the value x = 2 into the function and obtained arctan(0) as the result.