Full solution

Q. Evaluate the limit:

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

Identify indeterminate form: Identify the indeterminate form. $\newline$ We need to check if the expression $(x-1)/(1-\sqrt{x})$ results in an indeterminate form when $x$ approaches $1$ . $\newline$ Substitute $x = 1$ into the expression to see if it results in $0/0$ . $\newline$ $(1-1)/(1-\sqrt{1}) = 0/0$ . $\newline$ Since we get $0/0$ , it is an indeterminate form.
Simplify expression: Simplify the expression to resolve the indeterminate form. $\newline$ We can use algebraic manipulation to simplify the expression. One common technique is to multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ The conjugate of $(1-\sqrt{x})$ is $(1+\sqrt{x})$ . Multiply both the numerator and denominator by this conjugate. $\newline$ $\frac{(x-1)}{(1-\sqrt{x})} \cdot \frac{(1+\sqrt{x})}{(1+\sqrt{x})} = \frac{(x-1)\cdot(1+\sqrt{x})}{(1-\sqrt{x})\cdot(1+\sqrt{x})}$ .
Perform multiplication: Perform the multiplication in the numerator and denominator. $\newline$ Expand the numerator: $(x-1)\cdot(1+\sqrt{x}) = x + x\sqrt{x} - 1 - \sqrt{x}$ . $\newline$ Expand the denominator using the difference of squares: $(1-\sqrt{x})\cdot(1+\sqrt{x}) = 1 - (\sqrt{x})^2 = 1 - x$ . $\newline$ Now the expression is $\frac{x + x\sqrt{x} - 1 - \sqrt{x}}{1 - x}$ .
Further simplify expression: Simplify the expression further. Notice that the terms in the numerator can be rearranged to cancel out with the denominator. Rearrange the numerator: $(x - 1) + x\sqrt{x} - \sqrt{x} = (x - 1) - (\sqrt{x} - x\sqrt{x})$ . Now the expression is $\frac{(x - 1) - (\sqrt{x} - x\sqrt{x})}{(1 - x)}$ .
Factor out common terms: Factor out common terms to cancel them. $\newline$ Factor $(x - 1)$ out of the numerator: $(x - 1)(1 - \sqrt{x})$ . $\newline$ Now the expression is $\frac{(x - 1)(1 - \sqrt{x})}{(1 - x)}$ . $\newline$ Since $(x - 1)$ is equivalent to $-(1 - x)$ , we can cancel out $(1 - x)$ in the numerator and denominator. $\newline$ The expression simplifies to $-(1 - \sqrt{x})$ .
Take limit: Take the limit of the simplified expression as $x$ approaches $1$ . Now that we have a simplified expression, we can substitute $x = 1$ directly. $\lim_{x \to 1} -(1 - \sqrt{x}) = -(1 - \sqrt{1}) = -(1 - 1) = 0$ .