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What is the value of k for which the fol...

What is the value of k for which the following fucntion f(x) is continuous for all x?
`f(x)={{:((x^(2)-3x+2)/((x-1)^(2))","," for "xne1),(k","," for " x=1):}`

A

3

B

2

C

1

D

-1

Text Solution

Verified by Experts

The correct Answer is:
A
Let `f(x)={{:((x^(3)-3x+2)/((x-1)^(2))",",AAxne1),(k",",AAx-1):}`
and f(x) is continuous.
`:.underset(xto1)limf(x)=k`
`impliesunderset(xto1)lim(x^(3)-3x+2)/((x-1)^(2))=k`
`impliesk=underset(xto1)lim(3x^(2)-3)/(2(x-1))` [By L' Hospitals rule]
`impliesk=underset(xto1)lim(6x)/(2)` [By L'Hospital rule]
`impliesk=3`
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