If a function is continuous at a point a, what is lim_{x->a} f(x)?

If a function is continuous at a point a, what is lim_{x->a} f(x)?

• A. The limit equals f(a).
• B. The limit may not exist.
• C. The limit equals f'(a).
• D. The limit equals 0.

Answer: (A)

When a function is continuous at a point a, the value the function takes at a matches the value that nearby inputs approach. In other words, the limit as x approaches a exists and equals f(a). So lim_{x->a} f(x) is simply f(a). This makes sense because continuity means there’s no jump or hole at a—the function’s value at a is the same as the limiting value from either side. For example, with f(x) = x and a = 3, the limit as x approaches 3 is 3, which equals f(3). The other possibilities would require special circumstances (like the limit not existing, or relating to the derivative or to zero), but they aren’t guaranteed by continuity in general.

When a function is continuous at a point a, the value the function takes at a matches the value that nearby inputs approach. In other words, the limit as x approaches a exists and equals f(a). So lim_{x->a} f(x) is simply f(a). This makes sense because continuity means there’s no jump or hole at a—the function’s value at a is the same as the limiting value from either side. For example, with f(x) = x and a = 3, the limit as x approaches 3 is 3, which equals f(3). The other possibilities would require special circumstances (like the limit not existing, or relating to the derivative or to zero), but they aren’t guaranteed by continuity in general.