How do check the graph of a function has an infinite discontinuity at a point or not? (without the help of a graphing calculator)
example: Check whether y = (x^2-9)/(3x-9) has an infinite discontinuity at x = 3 or not
CAn you get rid of the denominator?
(x+3)(x-3)/3(x-3) in the limit at x=3, this becomes 2.
To check whether a graph has an infinite discontinuity at a specific point, you need to check the behavior of the function as it approaches that point from both sides.
In the given example, we want to determine if the function y = (x^2-9)/(3x-9) has an infinite discontinuity at x = 3.
Step 1: Start by simplifying the function as much as possible. In this case, we can factor the numerator and denominator:
y = (x+3)(x-3)/(3(x-3))
Step 2: Now, we can cancel out the common factor of (x-3):
y = (x+3)/3
Step 3: Take the limit of the function as x approaches 3 from both sides. Let's do this separately for the left-hand limit (from the negative side) and the right-hand limit (from the positive side).
Left-hand limit:
lim(x→3-) (x+3)/3
Substitute x = 3 - h, where h is a small positive number approaching zero:
lim(h→0-) ((3 - h) + 3)/3
lim(h→0-) (6 - h)/3
As h approaches zero from the negative side, the numerator approaches 6 and the denominator remains constant at 3. Therefore, the left-hand limit is 6/3 = 2.
Right-hand limit:
lim(x→3+) (x+3)/3
Substitute x = 3 + h, where h is a small positive number approaching zero:
lim(h→0+) ((3 + h) + 3)/3
lim(h→0+) (6 + h)/3
As h approaches zero from the positive side, the numerator approaches 6 and the denominator remains constant at 3. Therefore, the right-hand limit is 6/3 = 2.
Step 4: Compare the left-hand limit and the right-hand limit. If they are equal, then the limit as x approaches 3 exists. However, if they are different or if either one of them does not exist, then there is a discontinuity at x = 3.
In this case, the left-hand limit (2) is equal to the right-hand limit (2). Therefore, the limit as x approaches 3 exists, indicating that there is no infinite discontinuity at x = 3 for the function y = (x^2-9)/(3x-9).
Note: If the left-hand and right-hand limits were different or one of them did not exist (e.g., approaching different values or approaching positive or negative infinity), then there would be an infinite discontinuity at x = 3.