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Solve the problem. To what new value should f(1) be changed to remove the discontinuity? f(x)=⎩⎨⎧​x2+21x+2​x<1x=2x>1​ 4 2 1 3

1 Answer

7 votes

To remove the discontinuity at
\displaystyle\sf x=1, we need to find the new value
\displaystyle\sf f(1) should be assigned.

Given the function
\displaystyle\sf f(x):


\displaystyle\sf f(x)=\begin{cases}x^(2)+2, &amp; x<1\\2x, &amp; x=1\\3, &amp; x>1\end{cases}

To remove the discontinuity at
\displaystyle\sf x=1, we need to ensure that the left-hand limit and the right-hand limit of
\displaystyle\sf f(x) at
\displaystyle\sf x=1 are equal.

The left-hand limit is obtained by evaluating
\displaystyle\sf f(x) as
\displaystyle\sf x approaches
\displaystyle\sf 1 from the left:


\displaystyle\sf \lim_(x\to 1^(-))f(x)=\lim_(x\to 1^(-))(x^(2)+2)=(1^(2)+2)=3

The right-hand limit is obtained by evaluating
\displaystyle\sf f(x) as
\displaystyle\sf x approaches
\displaystyle\sf 1 from the right:


\displaystyle\sf \lim_(x\to 1^(+))f(x)=\lim_(x\to 1^(+))3=3

Since the left-hand limit and the right-hand limit are both equal to
\displaystyle\sf 3, we can assign
\displaystyle\sf f(1) the value of
\displaystyle\sf 3 to remove the discontinuity.

Therefore, the new value for
\displaystyle\sf f(1) should be
\displaystyle\sf 3.


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