Let` f(x)=(1-x(1+|1-x|))/(|1-x|)cos((1)/(1-x))" for "x ne1.` then

Options

A. `lim_(xto1^-)f(x)=0`
B. `lim_(xto1^-)f(x)` does not exist
C. `lim_(xto1^+)f(x)=0`
D. `lim_(xto1^+)`does not exist.

Answer

A, D

Explanation

We have ,
`f(x)={((1-x(1+1-x))/(1-x),cos((1)/(1-x))" for",xlt1),((1-x(1-(1-x)))/(1-x),cos((1)/(1-x))" for",xgt1):}`
`f(x)={((1-x(2-x))/(1-x),cos((1)/(1-x))" for",xlt1),((1-x^2)/(-(1-x)),cos((1)/(1-x))" for",xgt1):}`
`f(x)={((1-x),cos((1)/(1-x))" for ",xlt1),((-1-x),cos((1)/(1-x))" for ",xgt1):}`
` therefore_(xto1^-)f(x)=lim_(xto1)(1-x) cos((1)/(1-x))`
`=0xx` (An oscillating number between -1 and 1)
` =0`
`lim_(xto1^+)f(x)=lim_(xto1)-(1+x)cos((1)/(1-x))`
`=-2` (An oscillating number between -1 and 1)
`rArr lim_(xto1^+)f(x)` does not exist.