1 Multiple choice 1 point Let $$ f(x)=\left\{\begin{array}{cc}(x-2)^{2}+1 ; & x
eq 2 \ k ; & x=2\end{array} ight. $$ The value of the constant $$ k $$ that makes the function $$ f(x) $$ continuous at $$ x=2 $$ is....

Question

1 Multiple choice 1 point Let $$ f(x)=\left\{\begin{array}{cc}(x-2)^{2}+1 ; & x 
eq 2 \ k ; & x=2\end{array}ight. $$ The value of the constant $$ k $$ that makes the function $$ f(x) $$ continuous at $$ x=2 $$ is....

UpStudy AI · Accepted Answer

To ensure the function $$ f(x) $$ is continuous at $$ x = 2 $$, the value of $$ k $$ must equal the limit of $$ f(x) $$ as $$ x $$ approaches 2.

Given:
$$ f(x) = 
  \begin{cases} 
   (x-2)^2 + 1 & \text{if } x \neq 2 \
   k & \text{if } x = 2 
  \end{cases}
$$

Calculate the limit as $$ x $$ approaches 2:
$$ \lim_{x \to 2} (x-2)^2 + 1 = (2-2)^2 + 1 = 0 + 1 = 1 $$

For continuity at $$ x = 2 $$:
$$ k = \lim_{x \to 2} f(x) = 1 $$

**Answer:** 1
The value of $$ k $$ is 1.

1 Multiple choice 1 point
Let
The value of the constant that makes the
function continuous at is…

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