Find the limit of the function (x^2+2x-3)/|x-1| if x approaches 1 from the left.

Expert Answers
justaguide eNotes educator| Certified Educator

We have to find the value of (x^2+2x-3)/|x-1| as x approaches from the left.

As x approaches from the left x - 1 is always negative, so we have |x - 1| = (1 - x)

lim x--> 1 [ (x^2+2x-3)/(1 - x)]

lim x--> 1 [ (x^2+3x - x-3)/(1 - x)]

lim x--> 1 [ (x(x + 3) - 1(x + 3)/(1 - x)]

lim x--> 1 [ (x - 1)(x + 3)/(1 - x)]

lim x--> 1 [ -(x + 3)]

substitute x = 1

=> - 4

The required value of the limit is -4

giorgiana1976 | Student

We are searching for the limit of the function for the values of x that are smaller than 1.

So, x<1.

We'll subtract 1 both sides:

x - 1<0

If x - 1<0 => |x-1| = -(x-1)

We'll substitute the denominator by -(x-1) and we'll factorize the numerator:

lim (x-1)(x+3)/-(x-1)

We'll simplify and we'll obtain:

lim -(x+3)

We'll substitute x by 1 and we'll get:

lim -(x+3) = -(1+3) = -4

The limit of the function, when x approaches to 1 from the left, is lim (x^2+2x-3)/|x-1| = -4.