1) which of the following is not a condition of continuity for a function f(x) at point a? a) f(a) exists b)…

1) which of the following is not a condition of continuity for a function f(x) at point a? a) f(a) exists b) lim f(x) exists (x→a) c) lim f(x)=f(a) (x→a) d) lim f(x) exists (x→∞)

1) which of the following is not a condition of continuity for a function f(x) at point a? a) f(a) exists b) lim f(x) exists (x→a) c) lim f(x)=f(a) (x→a) d) lim f(x) exists (x→∞)

Answer

Explanation:

Step1: Recall continuity conditions

A function $f(x)$ is continuous at $x = a$ if $f(a)$ exists, $\lim_{x\rightarrow a}f(x)$ exists and $\lim_{x\rightarrow a}f(x)=f(a)$.

Step2: Analyze each option

  • Option a: $f(a)$ existing is a condition for continuity at $a$.
  • Option b: $\lim_{x\rightarrow a}f(x)$ existing is a condition for continuity at $a$.
  • Option c: $\lim_{x\rightarrow a}f(x)=f(a)$ is a condition for continuity at $a$.
  • Option d: $\lim_{x\rightarrow\infty}f(x)$ existing has nothing to do with the continuity of $f(x)$ at the point $x = a$.

Answer:

d) $\lim_{x\rightarrow\infty}f(x)$ exists