Limits of piecewise functions

[rberg1897]

Let

According to the definition of derivative, to compute , we need to compute the left-hand and right hand limits.

I think that the limits from both the left and right should be 0, but my online homework program is telling me this is incorrect. Any help?

[pka]

Let

According to the definition of derivative, to compute , we need to compute the left-hand and right hand limits. I think that the limits from both the left and right should be 0, but my online homework program is telling me this is incorrect. Any help?

[TEX]f'(0)[/TEX] does exist. What is its value?

[soroban]

Hello, rberg1897!

Let: [tex]f(x) \:=\:\begin{Bmatrix} \text{-}6x^2 + 5x & \text{ for }x \le 0 \\ 7x^2 + 5x & \text{ for } x > 0 \end{array}[/tex]

According to the definition of the derivative, to compute [tex]f'(0)[/tex],
. . we need to compute the left-hand and right-hand limits.

I think the limits from both the left and right should be 0, . . . . no
. . but my online homework program is telling me this is incorrect.

Did you find the derivatives?

[tex]f'(x) \:=\:\begin{Bmatrix} \text{-}12x + 5 & \text{ for }x \le 0 \\ 14x + 5 & \text{ for }x > 0 \end{array}[/tex]

[rberg1897]

I understand what the derivatives are and that the continuity of the function exists. I guess what I'm not understanding is what limit I'm supposed to find. I thought in a case like this that plugging 0 into the individual pieces of the function would give me the left and right hand limits, in which case the limits where the function meets at 0 would be equal at 0 and prove the continuity. What am I missing?

[tkhunny]

You do not seem to understand what you claim to understand. Please never, EVER substitute in this way until AFTER you have demonstrated continuity. Substitution to establish continuity is no good. It assumes what is to be proven.

[rberg1897]

Then what is the best way to prove continuity? I'm sorry, my instructor hasn't been a lot of help or I wouldn't be here.