In Exercises 67 - 70 , use the following function. f ( x ) = { 2 - x , x ≤ 1 ( x )

step 1 In this problem, we're going to take a look at the function F of X, and we're going to try to an

In Exercises 67 - 70 , use the following function. f ( x ) =...

In Exercises $67 - 70$ , use the following function. $f ( x ) = \left\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.$ Multiple Choice What is the value of $\lim _ { x \rightarrow 1 } f ( x ) ?$ $( \mathrm { A } ) 5 / 2 \quad ( \mathrm { B } ) 3 / 2$ (C) 1 (D) 0 (E) does not exist

In Exercises $67 - 70$ , use the following function.

$f ( x ) = \left\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.$

Multiple Choice What is the value of $\lim _ { x \rightarrow 1 } f ( x ) ?$

$( \mathrm { A } ) 5 / 2 \quad ( \mathrm { B } ) 3 / 2$ (C) 1 (D) 0 (E) does not exist

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Key Concepts

Piecewise Functions

Piecewise functions are defined by different expressions over various intervals of the input variable. This concept requires evaluating the appropriate expression based on the input value or limit approach, especially near boundary points where the definition of the function changes.

One-Sided Limits

One-sided limits involve finding the limit of a function as the input approaches a specific value from either the left or the right side. This concept is critical when dealing with piecewise functions, as the expressions on each side of the boundary may yield different behaviors.

Limit at a Point

The limit at a point is the value that a function approaches as the input gets arbitrarily close to that point from all directions. For functions with different definitions on different intervals, it is necessary to confirm that both the left-hand and right-hand limits are equal to determine if the limit at the point exists.

Continuity

Continuity at a point means that the function's value at that point matches the limit as the input approaches the point. In the context of piecewise functions, checking the equality of one-sided limits is essential to establish the overall continuity of the function at the boundary.

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