91Ó°ÊÓ

The given limits are

$$\begin{gathered} \underset{x→c}{\lim }f\left(x\right)=L \\ \underset{x→c^{-}}{\lim }f\left(x\right)=L \\ \underset{x→c^{+}}{\lim }f\left(x\right)=L \end{gathered}$$

Limit $\underset{x→c}{\lim }f\left(x\right)=L$state that if $x∈(c-\mathrm{δ},c)∪(c,c+\mathrm{δ}),$then $f\left(x\right)∈(L-\mathrm{Î\mu },L+\mathrm{Î\mu })$so that for all $\mathrm{Î\mu }>0,$there will be $\mathrm{δ}>0.$

for example in$\underset{x→3}{\lim }2x=6$, $x∈(3-\mathrm{δ},3)∪(3,3+\mathrm{δ})$and$2x∈(6-\mathrm{Î\mu },6+\mathrm{Î\mu })$

Left Limit $\underset{x→c}{\lim }f\left(x\right)=L$state that if role="math" localid="1648270910263" $x∈(c-\mathrm{δ},c),$then $f\left(x\right)∈(L-\mathrm{Î\mu },L+\mathrm{Î\mu })$so that for all $\mathrm{Î\mu }>0,$there will be $\mathrm{δ}>0.$

for example inrole="math" localid="1648271129853" $\underset{x→3^{-}}{\lim }2x=6$, $x∈(3-\mathrm{δ},3)$and$2x∈(6-\mathrm{Î\mu },6+\mathrm{Î\mu })$

Right Limit $\underset{x→c}{\lim }f\left(x\right)=L$state that if role="math" localid="1648270936304" $x∈(c+\mathrm{δ},c),$then $f\left(x\right)∈(L-\mathrm{Î\mu },L+\mathrm{Î\mu })$so that for all $\mathrm{Î\mu }>0,$there will be$\mathrm{δ}>0.$

for example inrole="math" localid="1648271285897" $\underset{x→3^{+}}{\lim }2x=6$,$x∈(3,3+\mathrm{δ})$and $2x∈(6-\mathrm{Î\mu },6+\mathrm{Î\mu })$