solve $$ \lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x}-1} $$

Question

UpStudy AI · Accepted Answer

To solve the limit

limₓ→1 (x - 1) / (√x - 1),

we first note that direct substitution gives 0/0, which is an indeterminate form.

A useful trick here is to express x - 1 in a way that cancels the problematic √x - 1 term. Notice that we can factor x - 1 as follows:

x - 1 = (√x)² - 1² = (√x - 1)(√x + 1).

So, we rewrite the expression:

(x - 1) / (√x - 1) = [(√x - 1)(√x + 1)] / (√x - 1).

For x ≠ 1, we can cancel the common factor (√x - 1):

= √x + 1.

Now, we can take the limit as x → 1:

limₓ→1 (√x + 1) = √1 + 1 = 1 + 1 = 2.

Thus, the limit is 2.
The limit is 2.

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