if lim f(x) exists with lim f(x) < 5 and f(6) = 10, which of the following statements must be false? a lim…

Explanation:

Step1: Recall the definition of continuity

For a function $y = f(x)$ to be continuous at $x = a$, $\lim_{x\rightarrow a}f(x)=f(a)$.

Step2: Analyze the given conditions

We are given that $\lim_{x\rightarrow 6}f(x)$ exists and $\lim_{x\rightarrow 6}f(x)<5$, while $f(6) = 10$. Since $\lim_{x\rightarrow 6}f(x)\neq f(6)$, the function cannot be continuous at $x = 6$.

Step3: Analyze other options

Option A: $\lim_{x\rightarrow 6}f(x)=0$ is possible as $0<5$. Option B: $\lim_{x\rightarrow 6^{+}}f(x)<5$ is consistent with the given $\lim_{x\rightarrow 6}f(x)<5$. Option C: Since $\lim_{x\rightarrow 6}f(x)$ exists, by the definition of the existence of a limit, $\lim_{x\rightarrow 6^{-}}f(x)=\lim_{x\rightarrow 6^{+}}f(x)$.

Answer:

D. $f$ is continuous at $x = 6$.