Question Number 66285 by Rio Michael last updated on 12/Aug/19
![What is the difference between lim_(x→2^− ) and lim_(x→2^+ )](https://www.tinkutara.com/question/Q66285.png)
$${What}\:{is}\:{the}\:{difference}\:{between} \\ $$$$\:\:\underset{{x}\rightarrow\mathrm{2}^{−} } {{lim}}\:\:{and} \\ $$$$\underset{{x}\rightarrow\mathrm{2}^{+} } {{lim}} \\ $$
Answered by MJS last updated on 12/Aug/19
![limit from left respectively right side. the two limits can be different lim_(x→2^− ) (1/(x−2)) =−∞ lim_(x→2^+ ) (1/(x−2))=+∞ lim_(x→0^− ) (x^2 )^(1/x) =+∞ lim_(x→0^+ ) (x^2 )^(1/x) =0](https://www.tinkutara.com/question/Q66286.png)
$$\mathrm{limit}\:\mathrm{from}\:\mathrm{left}\:\mathrm{respectively}\:\mathrm{right}\:\mathrm{side}.\:\mathrm{the} \\ $$$$\mathrm{two}\:\mathrm{limits}\:\mathrm{can}\:\mathrm{be}\:\mathrm{different} \\ $$$$\underset{{x}\rightarrow\mathrm{2}^{−} } {\mathrm{lim}}\:\frac{\mathrm{1}}{{x}−\mathrm{2}}\:=−\infty \\ $$$$\underset{{x}\rightarrow\mathrm{2}^{+} } {\mathrm{lim}}\frac{\mathrm{1}}{{x}−\mathrm{2}}=+\infty \\ $$$$ \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\sqrt[{{x}}]{{x}^{\mathrm{2}} }\:=+\infty \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\sqrt[{{x}}]{{x}^{\mathrm{2}} }\:=\mathrm{0} \\ $$
Answered by GordonYeeman last updated on 12/Aug/19
![the limit x→2 consists on taking arbitrarily close values of 2 (1.99, 1.9999,1.999999...). x→2^+ is basically the same but those values must be greater than 2 (2.01, 2.000001, etc). In the case of x→2^− they must be smaller](https://www.tinkutara.com/question/Q66288.png)
$${the}\:{limit}\:{x}\rightarrow\mathrm{2}\:{consists}\:{on}\:{taking}\: \\ $$$${arbitrarily}\:{close}\:{values}\:{of}\:\mathrm{2}\:\left(\mathrm{1}.\mathrm{99},\right. \\ $$$$\left.\mathrm{1}.\mathrm{9999},\mathrm{1}.\mathrm{999999}…\right).\:{x}\rightarrow\mathrm{2}^{+} \:{is}\:{basically} \\ $$$${the}\:{same}\:{but}\:{those}\:{values}\:{must}\:{be}\: \\ $$$${greater}\:{than}\:\mathrm{2}\:\left(\mathrm{2}.\mathrm{01},\:\mathrm{2}.\mathrm{000001},\:{etc}\right).\: \\ $$$${In}\:{the}\:{case}\:{of}\:{x}\rightarrow\mathrm{2}^{−} \:{they}\:{must}\:{be} \\ $$$${smaller} \\ $$
Commented by Rio Michael last updated on 12/Aug/19
![thank you sirs](https://www.tinkutara.com/question/Q66291.png)
$${thank}\:{you}\:{sirs} \\ $$