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:
is continuous.
has a jump discontinuity.
has an infinite discontinuity.
has a removable discontinuity.
None of these
how about you giving us your ideas on the rest of these? They're all quite similar, and none is particularly hard.
To determine the continuity of the function g(x), we need to check the behavior of the function at its points of discontinuity, which occur at x = 1.
For a function to be continuous at a point, the function value and the limit of the function as x approaches that point must be the same.
Let's calculate the limit of g(x) as x approaches 1 from the left (x < 1):
lim (x → 1-) g(x) = lim (x → 1-) 1/(x-2)
Since x approaches 1 from the left, the denominator of the fraction approaches -1 (1-2), which means the fraction approaches negative infinity.
Now, let's calculate the limit of g(x) as x approaches 1 from the right (x ≥ 1):
lim (x → 1+) g(x) = lim (x → 1+) 2x - 3
Here, as x approaches 1 from the right, the expression 2x - 3 approaches -1 (2(1) - 3).
Since the limits of g(x) do not agree, i.e., the limit from the left is negative infinity and the limit from the right is -1, g(x) has a jump discontinuity at x = 1.
Therefore, the best description concerning the continuity of g(x) is that the function has a jump discontinuity.