Calculus 2a?? Limits?

Question

A) lim( x > 1 ) (((x^3) - 1) / ((x^2) - 1))

B) lim( x > .5) ((2x) / ( x + (1/x)))

Answer 1

A. Notice that
(x^3 - 1)/(x^2 - 1) = ((x - 1)*(x^2 + x + 1))/((x - 1)*(x + 1). Now if x is not = 1, we can cancel the x - 1 factor from numerator and denominator. But we are assured that x is not 1 when taking a limit as x > 1 becasuse of the part of the definition that says 0 < abs(x - 1) < delta ... Notice that 0 < abs means x is not 1. Therefore the simplified expression is (x^2 + x + 1)/(x + 1). This is continuous at x = 1 so we can evaluate the limit by taking x = 1. The limit is 3/2.

B. This function is continuous at x = .5, so we can find the limit by evaluating at x = .5. The limit is 2/5.

Answer 2

A. lim x go to 1 (x^3-1)/((x^2-1) = lim[ (x-1)(x^2 + x+1)]/[(x-1)(x+1)]
= lim (x^2+x+1)/(x+1)
put 1 for x
lim (1^2+1+1)/(1+1)
lim (3/2)
3/2
B. put .5 for x
lim(2*.5/(.5+1/.5) = lim 1/(2.5) = .4