find the limit as x approaches 1 of (sin(x-1))/(x2+x-2)

Question

How do i find the limit as x approaches 1 of (sin(x-1))/(x^2 +x-2) for a calculus course at columbia homework problem?

Answer 1

As x approaches 1 the numerator approaches zero and the denominator approaches 2, so the quotient approaches 0/2=0.

If the denominator had been x^2-x^-2, the limit would be 1/4:

Let x-1=h where h is very small, the denominator would be:

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

(h^2+2h)(h^2+2h+2)/(h^2+2h+1)=4h where h is small enough to ignore h^2 and higher powers of h, and also ignore h because it's small compared to 1 or 2. Also, sin(h)=h for h small.

The quotient is therefore h/4h=1/4 as x approaches 1.