help hard calculus question

Question

sometimes, substituiting x=a into lim x->a f(x) can yield the interminate form 0/0.

consider the limit lim x->2 x^2-x-2/x-2.

a. explain why direct substitution does not work for this limit.

b. what is the equivalent function that is the same a x^2-x-2/x-2 for all values except at x=2?

c. determine the limit and explain why this method is acceptable.

d. explain what the graph will look like.

Answer 1

factor the numerator, then cancel the factor x-2

(x^2 -x -2)/(x-2) = (x-2)(x+1)/(x-2)= x+1

the graph of y = x+1 is a straight line, with slope 1 and y intercept 1

the original equation y = (x^2-x -1)/(x-1) is the identical straight line as y =x+1, but with a tiny "hole" in it, a discontinuity at x=2. the point (2,3) is not on the line. Draw the graph as a dark straight line, but with an open dot at (2,3) dividing the line into 2 separate lines.

the limit is found by plugging x=2 into the equation y=x+1