Question

$\mathrm{If}f\left(x\right)=\left\{\begin{array}{l}x,x>1\\ x^2,x<1\end{array}\right.,\mathrm{then}\underset{x\to 2}{\lim }f\left(x\right)=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

Answer 1

$$\begin{gathered} f\left(x\right)\left\{\begin{array}{l}x;x>1\\ x^2;x<1\end{array}\right. \\ \mathrm{Hence},\begin{array}{c}\lim \\ x\to 1\end{array}f\left(x\right)\mathrm{for}x<1 \\ \mathrm{i}.\mathrm{e}.\mathrm{L}.\mathrm{H}.\mathrm{L}\mathrm{is}\mathrm{given}\mathrm{by}\begin{array}{c}\lim \\ x\to 1^{-}\end{array}f\left(x\right) \\ =\begin{array}{c}\lim \\ x\to 1\end{array}{x}^2={\left(1\right)}^2=1 \\ \mathrm{i}.\mathrm{e}.\mathrm{L}.\mathrm{H}.\mathrm{L}\begin{array}{c}\lim \\ x\to 1^{-}\end{array}f\left(x\right)=1 \\ \mathrm{and}\mathrm{R}.\mathrm{H}.\mathrm{L}\begin{array}{c}\lim \\ x\to 1^{+}\end{array}f\left(x\right)=\begin{array}{c}\lim \\ x\to 1\end{array}x=1 \\ \mathrm{Hence}\mathrm{L}.\mathrm{H}.\mathrm{L}=\mathrm{R}.\mathrm{H}.\mathrm{L} \\ \mathrm{i}.\mathrm{e}.\begin{array}{c}\lim \\ x\to 1\end{array}f\left(x\right)=1 \end{gathered}$$