Question

Multiple Choice Which of the following statements about the function \(f(x)=\left\\{\begin{array}{ll}{2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\\ {-x+3,} & {1 < x < 2}\end{array}\right.\) is not true? (A) \(f(1)\) does not exist. (B) \(\lim _{x \rightarrow 0^{+}} f(x)\) exists. (C) \(\lim _{x \rightarrow 2^{-}} f(x)\) exists. (D) \(\lim _{x \rightarrow 1} f(x)\) exists. (E) \(\lim _{x \rightarrow 1} f(x)=f(1)\)

Short answer

The statements A and E are not true about this function.

Step-by-step solution

Analyzing statement A

Examine the value for \(f(1)\). It can be observed from the definition of the function that \(f(1) = 1\). Thus, statement A is not true as \(f(1)\) does exist and equals 1.

Analyzing statement B

Calculate \(\lim _{x \rightarrow 0^{+}} f(x)\). As x tends to 0 from the positive side, we're under the first condition of the piece-wise function, \(2x\). Hence, the limit is \(2 * 0 = 0\). This means statement B is true

Analyzing statement C

Calculate \(\lim _{x \rightarrow 2^{-}} f(x)\). As x tends to 2 from the negative side, we are in the last condition of the piece-wise function, \(-x + 3\). So, the limit is \(-2 + 3 = 1\), which implies that statement C is also true

Analyzing statement D

Calculate limit \(\lim _{x \rightarrow 1} f(x)\). To fully analyze, we consider limits from both sides of 1. The limit from the left side (\(x \rightarrow 1^{-}\)) is found under the first condition, \(2x\), giving us \(2 * 1 = 2\). However the limit from the right side (\(x \rightarrow 1^{+}\)) is under the last condition, \(-x + 3\), providing \(-1 +3 = 2\). Both sides provide the same limit, which means that the limit \(\lim _{x \rightarrow 1} f(x)\) does exist. Statement D is true.

Analyzing statement E

Check if \(\lim _{x \rightarrow 1} f(x) = f(1)\). From step 1, we know that \(f(1) = 1\). From step 4, we found that \(\lim _{x \rightarrow 1} f(x) = 2\). So, \(\lim _{x \rightarrow 1} f(x) \neq f(1)\) which implies that statement E is not true.