Limits — Question 2

Question 2 – Solution

Strategy. Factor the numerator to expose a common factor with the vanishing denominator.

Step 1: Indeterminate form At $(1,2)$, numerator and denominator are both $0$, so substitution gives $0/0$, not a limit value.

Step 2: Factor Treat the numerator as a polynomial in $y$: $$x^2-3x+2+y(x-1)=(x-1)(x-2)+y(x-1)=(x-1)(x+y-2).$$ Whenever $x\ne 1$, $$\frac{(x-1)(x+y-2)}{x-1}=x+y-2.$$

Step 3: Limit The simplified function is continuous, so $$\boxed{\displaystyle\lim_{(x,y)\to(1,2)}\frac{x^2+xy-3x-y+2}{x-1}=1}.$$ The original quotient is undefined on the entire line $x=1$, including the target, but points of its domain approach $(1,2)$ and the missing line does not change the limit.