Question Number 192999 by Mastermind last updated on 01/Jun/23
![Show that the following functions are continous on a close interval [0, 1]. f(x)={_(3 x=1) ^(((x^2 +x−2)/(x−1)) x≠1) Help!](https://www.tinkutara.com/question/Q192999.png)
$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{following}\:\mathrm{functions} \\ $$$$\mathrm{are}\:\mathrm{continous}\:\mathrm{on}\:\mathrm{a}\:\mathrm{close}\:\mathrm{interval} \\ $$$$\left[\mathrm{0},\:\mathrm{1}\right]. \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\left\{_{\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}=\mathrm{1}} ^{\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{x}−\mathrm{2}}{\mathrm{x}−\mathrm{1}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}\neq\mathrm{1}} \right. \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$
Answered by MM42 last updated on 02/Jun/23
![lim_(x→0^+ ) f(x)=f(0)=2 lim_(x→1^− ) f(x)=lim_(x→1^− ) (((x−1)(x+2))/(x−1))=3 & f(1)=3 ∀c∈(0,1) ; lim_(x→c) f(x)=f(c) therfore ; f is continous on [0,1]](https://www.tinkutara.com/question/Q193007.png)
$${lim}_{{x}\rightarrow\mathrm{0}^{+} } {f}\left({x}\right)={f}\left(\mathrm{0}\right)=\mathrm{2}\: \\ $$$${lim}_{{x}\rightarrow\mathrm{1}^{−} } {f}\left({x}\right)={lim}_{{x}\rightarrow\mathrm{1}^{−} } \frac{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)}{{x}−\mathrm{1}}=\mathrm{3}\:\:\&\:{f}\left(\mathrm{1}\right)=\mathrm{3}\: \\ $$$$\forall{c}\in\left(\mathrm{0},\mathrm{1}\right)\:;\:{lim}_{{x}\rightarrow{c}} {f}\left({x}\right)={f}\left({c}\right) \\ $$$${therfore}\:;\:{f}\:{is}\:{continous}\:{on}\:\left[\mathrm{0},\mathrm{1}\right]\: \\ $$