Explain why the function is discontinuous at the given number a. Sketch the graph of the function.f(x) = \left\{\frac{1}{x+2}\right\} if x \neq -2 a= -21 if x = -2

aflacatn 2021-05-17 Answered

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. \(f(x) = \left\{\frac{1}{x+2}\right\}\) if \(x \neq -2\) \(a= -2\)
1 if \(x = -2\)

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Demi-Leigh Barrera
Answered 2021-05-18 Author has 24846 answers
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content_user
Answered 2021-09-28 Author has 10829 answers

A function f is continuous from the right at x=a if

\(\lim_{x\to a^+}f(x)=f(a)\)

And f is continuous from the belt at x=a if

\(\lim_{x\to a^-}f(x)=f(a)\)

f is said to be continuous when it is continuous from both left and right

\(f(x)=\begin{cases}\frac{1}{x+2}&if\ x\ne-2\\1 &if\ x=-2\end{cases}\)

So we can see that:

\(\lim_{x\to-2^-}f(x)=-\infty\)

And

\(\lim_{x\to-2^-}f(x)=+\infty\)

Bat  \(f(-2)=1\)

Therefore fis discontinuous at x=-2 from both left and right

Result: f is discontinuous at x=-2 from both left and right

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