Home › JEE Advanced › Calculus › Which condition correctly states that a function…
Which condition correctly states that a function $f(x)$ is continuous at $x=0$?
A$\lim_{x\to 0^-}f(x)=\lim_{x\to 0^+}f(x)$ only
B$\lim_{x\to 0}f(x)$ exists, even if $f(0)$ is different
C$\lim_{x\to 0^-}f(x)=\lim_{x\to 0^+}f(x)=f(0)$
D$f(0)$ exists and both one-sided limits need not exist
Answer & Solution
Correct answer: C. $\lim_{x\to 0^-}f(x)=\lim_{x\to 0^+}f(x)=f(0)$
A function is continuous at a point when the left-hand limit, right-hand limit, and the function value at that point are all equal. Equality of only the one-sided limits is not enough unless that common value also equals $f(0)$.
Related questions
A tank is draining and a student is asked how fast the depth falls when the volume is chanThe notation used for a family of antiderivatives, complete with its constant, is the:Knowing an object's velocity, a student recovers its position by finding an:A function whose derivative is the given function is called its:Calculators and computers rely on that same tangent-based idea when they find:An iterative technique for finding zeroes, built on tangent line approximations, is:The rule that resolves such a limit by differentiating numerator and denominator separatelA limit produces the form zero over zero, so its behaviour cannot be read off directly. Th