Which of the following best describes the limit as x approaches 4 of the quotient of 2 times x divided by the quantity negative 2 plus square root of x ?


It exists and equals 4

It fails to exist because it is unbounded.

It fails to exist because its one-sided limits are not equal to the same number.

It fails to exist because it oscillates.

what is wrong with using symbols?

2x/(√x-2)
the limit as x->4 is 8/0 which is undefined.

In fact, since √x-2 is positive as x->4 from the right and negative as x->4 from the left, it's both unbounded and unequal from left and right.

I think (b) is the better answer, since the "value" from left and right are not really a number; they are undefined.

To find the limit, we need to evaluate the expression as x approaches 4. Let's simplify the expression first.

The given expression is: (2x) / (-2 + √x)

As x approaches 4, we substitute x = 4 into the expression:

(2 * 4) / (-2 + √4)

This simplifies to:

8 / (-2 + 2)

Since (-2 + 2) equals 0, the expression becomes:

8 / 0

When the denominator of a fraction is zero, the fraction is considered undefined. Therefore, the limit fails to exist because the denominator is zero.

So, the correct answer is: "It fails to exist because its one-sided limits are not equal to the same number."