The value of k

Question

If , what must be the value of k in order for f to be continuous?

–2

–1

1

2

Answer 1

A function is continuous if, for all a, the limit as x approaches a from both directions is the same as f(a). This is trivially true for all values of a except for 1; it is only at x=1 that there is a potential problem, and so there we will focus our efforts.

As we approach 1 from the left and right, the function is described by [(1/x)-1]/[x-1], and so we need to find the limit as x approaches 1 of this quantity. We find that

lim (x->1) [(1/x)-1]/[x-1]

=lim (x->1) [1-x]/[x(x-1)] Multiply the top and bottom by x

=lim (x->1) [(-1)(1-x)]/[x(1-x)] Multiply the top and bottom by -1

=lim (x->1) -1/x

=-1

So, the limit as x approaches 1 for this function is -1. So, we need f(1)=-1. That means that

f(1)=1-k=-1

And so k=2.