Question

To this function, does limit x approaching zero (without + or -) exist? Aside from that, is f(x) continuous at x=1?

Answer 1

Given:

`f(x)=\begin{cases}{-(1)/(x),x<0} \\ {3,0\leq x<1} \\ {√(x)}+2,x\ge1\end{cases}`

Required:

To find the limit exist if x approaching to 0.

Step-by-step explanation:

As limit x approaching to 0,

`\lim_(x\rightarrow0)f(x)=3`

Therefore the limit exist.

`\begin{gathered} \lim_(x\rightarrow1^(-1))f(x)=\lim_(x\rightarrow1^(-1))3 \\ \\ =3 \end{gathered}`
`\begin{gathered} \lim_(x\rightarrow1^+)f(x)=\lim_(x\rightarrow1^+)√(x)+2 \\ \\ =√(1)+2 \\ \\ =1+2 \\ \\ =3 \end{gathered}`