Let f(x)= (x^2-1)/(x-1). A) find I) lim f(x) x—>1+ II) lim f(x) x—>1-

Question

does Lim x—> 1 f(x) exist? 

Sketch the graph of f

Answer 1

f(x) = (x²-1)/(x-1),   x≠1

 

Now x²-1 = (x+1)(x-1)

 

f(x) = (x+1)(x-1)/(x-1) = x+1   (but x≠1)

 

lim f(x) as x-->1+ = 1+1 = 2

lim f(x) as x-->1-  = 1+1 = 2

 

Since limits from both sides are equal, the limit exists and it equals 2.  The graph of f(x) = x+1 is a straight line with a slope of 1 and y-intercept of 1.  There is a "hole" at x=1, meaning they-value at x=1 is undefined.  The limit of f(x) at x=1 does exist, however, and equals 2.