What is the limit, as x approaches 1, of (sqrt(x) - 1)/(x - 1)?
I need to show work, but I know the answer is 3/2, because I confirmed with a TI-89.
perhaps you have confused the answer with that of a different problem, the answer is not 3/2.
Since putting in x=1 gives 0/0, you can either use d'Hôpital's rule, or simply multiply by the conjugate (sqrt(x)+1) of the numerator to create the common factor (x-1).
So
(sqrt(x) - 1)/(x - 1)
=(sqrt(x)-1)(sqrt(x)+1)/[(x-1)(sqrt(x)+1)]
=(x-1)/[(x-1)(sqrt(x)+1)]
=1/(sqrt(x)+1)
=1/(sqrt(1)+1)
=1/2
Ah, the good old limit of (sqrt(x) - 1)/(x - 1) as x approaches 1! Let's get to the bottom of it, shall we?
First, let's simplify the expression a bit. Applying the difference of squares, we can rewrite it as [(sqrt(x) - 1)(sqrt(x) + 1)] / [(x - 1)(sqrt(x) + 1)]. Now, we can cancel out the (sqrt(x) + 1) terms in the numerator and denominator. This leaves us with (sqrt(x) - 1) / (x - 1).
Since we're taking the limit as x approaches 1, we can't just substitute the value directly. Instead, let's focus on the numerator. We know that as x approaches 1, the numerator approaches (sqrt(1) - 1), which is 0.
Now, let's examine the denominator. Again, as x approaches 1, the denominator approaches (1 - 1), which is 0.
So, to answer your question, as x approaches 1, the expression (sqrt(x) - 1)/(x - 1) is indeterminate. It's not in the form of an actual number like 3/2. Sorry to rain on your Ti-89 parade!
Remember, though, this is no reason to feel down. The beauty of limits is that they help us understand how functions behave near a certain point. So while we can't assign a single value to this limit, we can analyze the behavior of the expression as x gets infinitely close to 1.
To find the limit, as x approaches 1, of the expression:
lim (x -> 1) [(sqrt(x) - 1)/(x - 1)]
We can start by simplifying the expression. The numerator can be simplified using the conjugate:
(sqrt(x) - 1)(sqrt(x) + 1) = (x - 1)
So the limit becomes:
lim (x -> 1) [(x - 1)/(x - 1)]
Notice that both the numerator and denominator are equal when x = 1, except when x = 1. Therefore, we have an indeterminate form (0/0) which requires further simplifying. We can cancel out the common factor:
lim (x -> 1) [1] = 1
Thus, the limit as x approaches 1 of the expression is equal to 1.
The result you stated (3/2) is incorrect. It appears to be a calculation error or confusion with a different problem. The correct answer for this particular limit is 1.
To find the limit of the given expression as x approaches 1, we can simplify the expression by rationalizing the numerator. Here's the step-by-step solution:
1. Start with the given expression:
lim(x → 1) [(√x - 1)/(x - 1)]
2. Rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator, which is (√x + 1):
lim(x → 1) [(√x - 1)/(x - 1)] * [(√x + 1)/(√x + 1)]
Simplifying the numerator gives: lim(x → 1) [(√x)^2 - 1] = lim(x → 1) (x - 1)
3. Now, the expression becomes:
lim(x → 1) (x - 1)/(x - 1)
4. We can cancel out the common factor of (x - 1):
lim(x → 1) 1
5. The limit as x approaches 1 of a constant function is just the constant value itself. Therefore, the final result is:
1
So, the limit, as x approaches 1, of (sqrt(x) - 1)/(x - 1) is 1, not 3/2 as you mentioned. It's important to double-check computational results and verify them using proper mathematical methods.