The limit type for a function where lim_{x->a} f(x) exists but f(a) ≠ lim_{x->a} f(x) is what?

The limit type for a function where lim_{x->a} f(x) exists but f(a) ≠ lim_{x->a} f(x) is what?

• A. Jump Discontinuity
• B. Infinite Discontinuity
• C. Continuous
• D. Removable Discontinuity

Answer: (D)

When x approaches a, the function settles to a finite value, but the value at a is not that same limit. That creates a hole in the graph at a: the limit exists, yet f(a) differs from it. This situation is a removable discontinuity, because you could redefine f(a) to be the limit value and make the function continuous at a. The other types don’t fit: a jump discontinuity would require the left- and right-hand limits to be different, so the overall limit wouldn’t exist. An infinite discontinuity involves the function growing without bound near a, so the limit isn’t finite. If the function value matched the limit, it would be continuous.

When x approaches a, the function settles to a finite value, but the value at a is not that same limit. That creates a hole in the graph at a: the limit exists, yet f(a) differs from it. This situation is a removable discontinuity, because you could redefine f(a) to be the limit value and make the function continuous at a.

The other types don’t fit: a jump discontinuity would require the left- and right-hand limits to be different, so the overall limit wouldn’t exist. An infinite discontinuity involves the function growing without bound near a, so the limit isn’t finite. If the function value matched the limit, it would be continuous.