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I have this piecewise function:

$f(x) = \begin{cases} x^2 -5x+6, x\leq1 \\ ax+b, x > 1\end{cases}$

And i need to find the values $a,b$ such that $f$ is differentiable and continuous at $x=1$, using derivative definition.

My development was:

If $\lim_{h\to0} \frac{f(1+h)-f (1)}{h}$ exist, therefore is differentiable and continuos at $x=1$

Solving the lateral limits,

$\large{\lim_{h\to0^{-}} \frac{h^2+2h+1-5h-5+4}{h} = -3}$

$\large{\lim_{h\to0^{+}}\frac{a+b-2+ah}{h}}$ and this be equal to $-3$, that is:

$\large{\lim_{h\to0^{+}}\frac{a+b-2+ah}{h} = -3 = \lim_{h\to0}-3}$

$(\star) \frac{a+b-2+ah}{h}=-3 \iff a+b-2=-h(a+3)$ and since $h\to 0$, i have $a+b=2$ and if $a+b=2$, the limits becomes:

$(\star)\large{\lim_{h\to0^{+}}\frac{a+b-2+ah}{h}} = \large{\lim_{h\to0^{+}}a} = -3 \iff a = -3$ and hence $b = 5$

The $(\star)$ steps is where I don't know if my steps are valid. And how i can know if these are the unique values of $a,b$ such that the condition holds?

Thanks in advance.

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The first $(\star)$ is not correct. You're not after $\frac{a+b-2+ah}h=-3$; you're after$$\lim_{h\to0^+}\frac{a+b-2+ah}h=-3.\tag1$$Since $\lim_{h\to0^+}h=0$, in order that you have $(3)$, you must have $\lim_{h\to0^+}a+b-2+ah=0$, which is equivalent to $a+b=2$. And, if this condition holds, then the limit $(1)$ is equal to $a$. So, $a=-3$.

The rest is fine.

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  • $\begingroup$ Can you explain why i must have $ \lim_{h\to0^+}a+b-2+ah=0$ $\endgroup$ – Eduardo Sebastian Jun 9 '20 at 2:32
  • $\begingroup$ Because if the limit of a quotient exists (in $\Bbb R$) and the limit of the denominator is $0$, then the limit of the numerator must be $0$ too. $\endgroup$ – José Carlos Santos Jun 9 '20 at 5:47
  • $\begingroup$ I think that you mean that if the denominator tends to $0$ the numerator must tends to $0$, right? $\endgroup$ – Eduardo Sebastian Jun 9 '20 at 13:00
  • $\begingroup$ Yes, that is correct. $\endgroup$ – José Carlos Santos Jun 9 '20 at 13:05
  • $\begingroup$ Thanks for your help José. $\endgroup$ – Eduardo Sebastian Jun 9 '20 at 13:07

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