Questions: Let F(x)= (x-2)^2+1 for x<2, sin(x-2) for 2 ≤ x. Then lim (x → 2-) F(x)=

Transcript text: Let $F(x)=\left\{\begin{array}{cl}(x-2)^{2}+1 & x<2 \\ \sin (x-2) & 2 \leq x\end{array}\right.$ Then $\lim _{x \rightarrow 2^{-}} F(x)=$

Solution

Solution Steps

Step 1: Identify the Relevant Function Piece

To find $\lim _{x \rightarrow 2^{-}} F(x)$, we focus on the piece of the function defined for $x < 2$, which is given by: \[ F(x) = (x - 2)^2 + 1 \]

Step 2: Evaluate the Limit

We substitute $x = 2$ into the expression $(x - 2)^2 + 1$ to find the limit as $x$ approaches 2 from the left: \[ \lim _{x \rightarrow 2^{-}} F(x) = (2 - 2)^2 + 1 = 0 + 1 = 1 \]

Final Answer

The limit is $\boxed{1}$.

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