Limits Question: lim[x->1] [(2x - 3)(sqrt[x] - 1)] / [2x^2 + x - 3]

[rooks]

So the problem is with a limit question:

. . . . .\(\displaystyle \displaystyle \lim_{x \rightarrow 1}\, \left(\dfrac{(2x\, -\, 3)\left(\sqrt{\strut x\,}\, -\, 1\right)}{2x^2\, +\, x\, -\, 3}\right)\)

My attempt at a solution:

-Factor the denominator inside the parentheses to (2x+3)(x-1)

Then I was stuck.

The problem is that the denominator has a zero of 1, meaning that I cannot plug 1 in for x, else it returns undefined.

How can I evaluate the limit with this problem?

 

[Deleted member 4993]

So the problem is with a limit question:

. . . . .\(\displaystyle \displaystyle \lim_{x \rightarrow 1}\, \left(\dfrac{(2x\, -\, 3)\left(\sqrt{\strut x\,}\, -\, 1\right)}{2x^2\, +\, x\, -\, 3}\right)\)

My attempt at a solution:

-Factor the denominator inside the parentheses to (2x+3)(x-1)

Then I was stuck.

The problem is that the denominator has a zero of 1, meaning that I cannot plug 1 in for x, else it returns undefined.

How can I evaluate the limit with this problem?

Did you eliminate the common factor → (√x -1) - from the numerator and the denominator?

 

[mmm4444bot]

By the product rule, we can write the given limit as:

lim[x→1] (2x - 3) * lim[x→1] (√x - 1)/(2x^2 + x - 3)

Regarding the second limit above, start by rationalizing the numerator.

Multiply out the top only.

Then, use your factorization of 2x^2+x-3, to simplify (hint: x-1 cancels). :cool: