Question

For which of the following does lim f(x) as x approaches c exist?

1) The function f(x) is continuous at x = c
2) The function f(x) has a removable discontinuity at x = c
3) The function f(x) has a jump discontinuity at x = c
4) The function f(x) has an infinite discontinuity at x = c

Answer 1

The correct answer is option 1) The function f(x) is continuous at x = c, the limit of f(x) as x approaches c does exist.

Step-by-step explanation:

The limit of a function as x approaches a specific value, c, exists if the function is continuous at x = c. For a removable discontinuity, the limit may or may not exist, while for a jump discontinuity or an infinite discontinuity, the limit does not exist.

For option 2) The function f(x) has a removable discontinuity at x = c, the limit may or may not exist. A removable discontinuity occurs when there is a hole in the graph of the function at x = c, but the limit can still exist if the values around the hole approach a specific value.

For option 3) The function f(x) has a jump discontinuity at x = c, the limit does not exist. A jump discontinuity occurs when there is a vertical jump in the graph of the function at x = c, and the values from the left and right of the jump approach different values.

For option 4) The function f(x) has an infinite discontinuity at x = c, the limit does not exist. An infinite discontinuity occurs when the function approaches positive or negative infinity at x = c, and the limit is undefined.