Suppose g(x) = { 1 / (x-2) if x < 1
{2x - 3 if x >/= 1
The best description concerning the continuity of g(x) is that the function
A.) is continuous
B.) has a jump discontuity
C.) has an infinite discontuity
D.) has a removable discontuity
E.) None of the above
really?
lim(x→1-) = -1
lim(x→1+) = -1
so g(x) is continuous at x = 1
Unfortunately, that pesky 1/(x-2) is a problem at x=2
So C is correct
I choose E. None of the above.
Thanks you
To determine the continuity of the function g(x), we need to determine if there are any discontinuities.
Let's analyze the function g(x) based on the given conditions:
For x < 1, the function is defined as g(x) = 1 / (x - 2).
For x ≥ 1, the function is defined as g(x) = 2x - 3.
To determine the continuity at x = 1, we need to evaluate the limits from both directions:
1. Left-hand limit: lim(x→1-) g(x) = lim(x→1-) (1 / (x - 2))
When x approaches 1 from the left (x < 1), the denominator (x - 2) approaches -1, and 1 / (-1) is -1.
2. Right-hand limit: lim(x→1+) g(x) = lim(x→1+) (2x - 3)
When x approaches 1 from the right (x ≥ 1), the value of 2x - 3 approaches -1.
Since both the left-hand and right-hand limits are equal at x = 1 (-1), the function is continuous at x = 1.
Therefore, the best description concerning the continuity of g(x) is A.) is continuous.