Question

1. \lim\limits_{x\to 1} \dfrac{x - 1}{\sqrt{x} - 1}

Explanation:

Step1: Rationalize the denominator

We can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $\sqrt{x} + 1$. So we have:

$$\lim_{x ightarrow1}\frac{x - 1}{\sqrt{x}-1}\times\frac{\sqrt{x}+1}{\sqrt{x}+1}=\lim_{x ightarrow1}\frac{(x - 1)(\sqrt{x}+1)}{x - 1}$$

Step2: Cancel out the common factor

Notice that $x - 1$ is a common factor in the numerator and the denominator (for $x
eq1$), so we can cancel them out:

$$\lim_{x ightarrow1}(\sqrt{x}+1)$$

Step3: Substitute the limit value

Now we substitute $x = 1$ into the expression $\sqrt{x}+1$:
$$\sqrt{1}+1=1 + 1=2$$

Answer:

2