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$17-20$ Explain why the function is discontinuous at the givennumber $a$ . Sketch the graph of the function.$f(x)=\left\{\begin{array}{ll}{e^{x}} & {\text { if } x<0} \\ {x^{2}} & {\text { if } x \geq 0}\end{array}\right. \quad a=0$

$\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x)$ and hence the function is not continuous at $a=0$

Calculus 1 / AB

Chapter 2

Limits

Section 5

Continuity

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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Lucy, it's one. Ask you to explain why dysfunction f is this continuous at point X equals zero? Well, to see that, first of all, we need to see what happened when the limit on the left. Okay, so this is love. Going to the left of zero to the left. Meanings their negative less going from something smaller to 0 to 0 up the function. So because we're looking at the to the left to the left, meaning smaller, smaller, then zero. So what function that you using using the blue guy here e to the x Andi that me and you plug in the zero there, and that would give you one. So didn't limit on the left is one now 40 other pen just doing the right limited. Right. So that would be a limit ex going to zero. But from the rice that's a plus. Baker than zero bigger than zero. Function half of backs and wished function. Are you using? You are going to you because x bigger and zero s. So you're using X square, right? There's an X square so that plucking zero square and that would be zero. So you see that because Lim from the left is not equal to two. So I left. It's not equal to the right. Therefore, you having a discontinuity discontinuity? And that's the reason why. All right, Uh, it's also asked us to kind of sketch the craft of the function so growing into the right here so you could see you, um, zero from the left. It is tthe e e to the X on. Using green again is something kind. This one didn't have a value at one. So the whole right there. Why e t the x for the blues? I am saying and then under right. It is ex squares and it has a value zero there. So that is not a hole. But this is a whole damn. And it is x. Where so something like that what you call x square. So the blue and a red together is the ground of this function. All right, See you in the next video.

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