Question

I have a calculus question about limits of a function, pic included

Answer 1

We are asked to find the left and right hand limits of the given function.

1. Left-hand limit:

`\lim_(n\to1^-)\;(|7x-7|)/(1-x)`

The numerator is negative when we substitute -1, so the absolute value will be evaluated as a negative expression.

`\lim_(n\to1^-)\;(-(7x-7))/(1-x)=(-7x+7)/(1-x)`

Now, factor the numerator

`\lim_(n\to1^-)\;(-7x+7)/(1-x)=(7(-x+1))/(1-x)=(7(1-x))/(1-x)=7`

So, the left-hand limit is equal to 7

2. Right-hand limit:

`\lim_(n\to1+)\;(|7x-7|)/(1-x)`

The numerator is positive when we substitute +1, so the absolute value will be evaluated as a positive expression.

`\lim_(n\to1+)\;(|7x-7|)/(1-x)=(7x-7)/(1-x)`

Now, factor the numerator and the denominator.

`\lim_(n\to1+)\;(7x-7)/(1-x)=(7(x-1))/(1-x)=(7(x-1))/(-(x-1))=-7`

So, the right-hand limit is equal to -7