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A function $f$ is said to have a removable discontinuity at $x=c$ if lim $_{x \rightarrow c} f(x)$ exists but $f$ is not continuous at $x=c$, either because $f$ is not defined at $c$ or because the definition for $f(c)$ differs from the value of the limit. This terminology will be needed in these exercises.(a) Sketch the graph of a function with a removable discontinuity at $x=c$ for which $f(c)$ is undefined.(b) Sketch the graph of a function with a removable discontinuity at $x=c$ for which $f(c)$ is defined.

a. function $f(x)$ is undefined at $x=c,$ so the graph of the function has hole at $x=c$.b. $f(x)=2$ for $x=1,$ which has removable discontinuous at $x=1$ By the definition of the function $f(1)=2,$ but is not equal to $\lim _{x \rightarrow 1} 3 x+1=4$

Calculus 1 / AB

Chapter 1

LIMITS AND CONTINUITY

Section 5

Continuity

Functions

Limits

Continuous Functions

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we've been given a definition of a removable discontinuity which basically says that a ruble discontinuity is such that the limit exists as X approaches see. But that limit is not equal to the function antsy in part A. We specifically wanted to draw a graph where we have a removable discontinuity, but the function isn't defined at sea, so we need the limits as X approaches see to exist. So from both sides, it's gotta be approaching some function value. However, when ecstasy, there is no defined point for B. We want there to be a removable discontinuity at sea and the function ISI exists. So we've got the limit existing. Now we need the function to exist, but it can't be the same as the limits. So as long as we put a function at sea it any point except that hollow dot we have satisfied that condition

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