Suppose
g(x)={x^2+2x+1/x+1 if x<1
{2x 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.
has both infinite and removable discontinuities.
Was removable correct?
removable at x=1
To determine the continuity of the function g(x), we need to check if it is continuous at two points: x = 1 (where the two pieces of the function intersect) and x = -1 (where there is a possible issue with the denominator).
At x = 1, we need to check if the limit of g(x) as x approaches 1 from the left is equal to the limit of g(x) as x approaches 1 from the right.
From the left (x < 1), g(x) = x^2+2x+1 / (x+1).
Taking the limit as x approaches 1 from the left gives us:
lim (x→1-) (x^2+2x+1) / (x+1) = 4/2 = 2.
From the right (x ≥ 1), g(x) = 2x.
Taking the limit as x approaches 1 from the right gives us:
lim (x→1+) 2x = 2.
Since both limits are equal to 2, the function is continuous at x = 1.
At x = -1, we need to check if the function is defined and continuous.
For x < 1, g(x) is defined as x^2+2x+1 / (x+1), and at x = -1, the denominator becomes 0.
Therefore, there is a removable discontinuity at x = -1 because we can redefine g(x) to be continuous at that point.
In conclusion, the function g(x) has a removable discontinuity at x = -1 and is continuous everywhere else.
Therefore, the best description concerning the continuity of g(x) is that the function has a removable discontinuity.
To determine the continuity of the function g(x), we need to analyze the behavior of the function at x=1.
For continuity, three conditions must be met:
1. The function should be defined at x=1.
2. The limit of the function as x approaches 1 from both sides must exist.
3. The value of the function as x approaches 1 from both sides must equal the limit.
Let's analyze each condition:
1. The function is defined at x=1 because g(x) is defined as 2x for x ≥ 1, and we can substitute x=1 into this expression, resulting in g(1) = 2(1) = 2.
2. The limit of g(x) as x approaches 1 from both sides must exist:
- As x approaches 1 from the left (x < 1), g(x) becomes (x^2 + 2x + 1) / (x + 1). Plugging in x=1, we get (1^2 + 2(1) + 1) / (1 + 1) = 4/2 = 2.
- As x approaches 1 from the right (x ≥ 1), g(x) is simply 2x. Plugging in x=1, we get 2(1) = 2.
3. The value of the function at x=1 from both sides equals the limit:
g(1) = 2 (as calculated earlier)
The limit from the left is 2 (as calculated earlier)
The limit from the right is 2 as well.
Since all three conditions are met, the function g(x) is continuous at x=1. Therefore, the best description concerning the continuity of g(x) is that the function is continuous.