Problem 40
We're approaching x = 1 from the right side or the positive side. This means we start with a value like x = 1.5 and move closer to x = 1
Since 1.5 is larger than 1, we use the rule that goes with x > 1 which is the second part of the piecewise f(x) function.
Quite simply, we plug x = 1 into the second piece to get
f(x) = 3x-1
f(1) = 3(1)-1
f(1) = 2
As x approaches 1 from the right, we have the limit of f(x) approach 2 from above.
Answer: 2
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Problem 41
This time we're approaching from the left, or the negative side
So we could start with something like x = 0 and get closer to x = 1
We'll go with the rule that has x ≤ 1 this time.
f(x) = x^2+1
f(1) = 1^2+1
f(1) = 2
In this case, both pieces produce the same output when plugging in x = 1. Therefore, the function is continuous at x = 1
Answer: 2
