lim sqrt(x+2-1)/(x-1)

Substitute $x=1$ into the numerator and the denominator:
Numerator: $\sqrt{1+2}-1 = \sqrt{3}-1$
Denominator: $1-1 = 0$
The expression takes the form $\frac{\sqrt{3}-1}{0}$. Since $\sqrt{3} \approx 1.732$, the numerator $\sqrt{3}-1 \approx 0.732$, which is a positive non-zero value.

Since the numerator approaches a non-zero value and the denominator approaches zero, the limit will be either $+\infty$, $-\infty$, or it will not exist. We need to examine the sign of the denominator as $x$ approaches 1 from both sides.

• As $x \to 1^+$ (x approaches 1 from values greater than 1): The denominator $x-1$ will be positive. The numerator $\sqrt{x+2}-1$ is positive (as $\sqrt{3}-1 > 0$). So, $\lim_{x \to 1^+} \frac{\sqrt{x+2}-1}{x-1} = \frac{\text{positive}}{\text{positive}} = +\infty$.

• As $x \to 1^-$ (x approaches 1 from values less than 1): The denominator $x-1$ will be negative. The numerator $\sqrt{x+2}-1$ is positive. So, $\lim_{x \to 1^-} \frac{\sqrt{x+2}-1}{x-1} = \frac{\text{positive}}{\text{negative}} = -\infty$.

Since the left-hand limit ($-\infty$) and the right-hand limit ($+\infty$) are not equal, the overall limit does not exist.