Evaluate the limit, if it exists.

Question

lim x->0 ((9/x)-(9/(x^2-x)))

Answer 1

To evaluate this limit, we can use limit laws and algebraic manipulation.

lim x->0 ((9/x)-(9/(x^2-x)))

First, let's simplify the expression by finding a common denominator for the fractions in the numerator:

lim x->0 ((9(x^2-x)/(x(x^2-x))) - (9/(x^2-x)))

Now, let's combine the fractions in the numerator:

lim x->0 ((9(x^2-x) - 9)/(x(x^2-x)))

Next, we can simplify the numerator:

lim x->0 (9x^2 - 9x - 9)/(x(x^2-x))

Now we can factor out 9 from the numerator:

lim x->0 (9(x^2 - x - 1))/(x(x^2-x))

Next, we can cancel out the common factors of x in the denominator and numerator:

lim x->0 (9(x - 1 - 1/x))/(x(x - 1))

Simplifying further, we get:

lim x->0 (9(x - 1 - 1/x))/(x^2 - x)

Now, before we proceed, we need to check if there are any cancellations or factors that make the function undefined at x = 0. We can see that the function is defined for all values of x except x = 0.

Now, let's continue:

lim x->0 (9(x - 1 - 1/x))/(x^2 - x)

Using limit laws, we can split the limit into two separate limits:

lim x->0 9(x - 1 - 1/x) / lim x->0 (x^2 - x)

Taking the limits separately:

lim x->0 9(x - 1 - 1/x)

Since we have a difference of fractions, we can combine the terms:

lim x->0 (9x^2 - 9x - 9)/(x(x^2-x))

Now, let's plug in x = 0:

lim x->0 (9(0) - 9(0) - 9)/(0(0^2-0))

This simplifies to:

lim x->0 (-9)/(0)

However, this is undefined because we have a division by zero. Therefore, the limit does not exist.