Which one of the following functions is continuous at \(x\) = 3?
2013
Which one of the following functions is continuous at \(x\) = 3?
- A.
\(f(x) = \begin{cases} 2,&\text{if $x = 3$ } \\ x-1& \text{if $x > 3$}\\ \frac{x+3}{3}&\text{if $x < 3$ } \end{cases}\) - B.
\(f(x) = \begin{cases} 4,&\text{if $x = 3$ } \\ 8-x& \text{if $x \neq 3$} \end{cases}\) - C.
\(f(x) = \begin{cases} x+3,&\text{if $x \leq 3$ } \\ x-4& \text{if $x > 3$} \end{cases}\) - D.
\(f(x) = \begin{cases} \frac{1}{x^3-27}&\text{if $x \neq 3$ } \end{cases}\)
Attempted by 47 students.
Show answer & explanation
Correct answer: A
How to check continuity at x = 3:
Compute the left-hand limit using f(x) = (x+3)/3 for x < 3: (3+3)/3 = 2.
Compute the right-hand limit using f(x) = x − 1 for x > 3: 3 − 1 = 2.
Evaluate the function at x = 3: f(3) = 2.
Conclusion: left-hand limit = right-hand limit = function value = 2, so the function is continuous at x = 3.