1. **State the problem:** Find the limit $$\lim_{x \to 1} \frac{\frac{2}{x} + 3 + x - 1}{x - 5} \div (x - 1).$$ 2. **Rewrite the expression:** The expression can be written as $$\lim_{x \to 1} \frac{\frac{2}{x} + 3 + x - 1}{(x - 5)(x - 1)}.$$ 3. **Simplify the numerator:** Combine like terms in the numerator: $$\frac{2}{x} + 3 + x - 1 = \frac{2}{x} + (3 - 1) + x = \frac{2}{x} + 2 + x.$$ 4. **Substitute $x=1$ directly:** $$\frac{2}{1} + 2 + 1 = 2 + 2 + 1 = 5,$$ and the denominator at $x=1$ is $$(1 - 5)(1 - 1) = (-4)(0) = 0,$$ which means direct substitution leads to division by zero. 5. **Check if numerator also approaches zero:** Numerator at $x=1$ is 5, not zero, so the limit tends to infinity or does not exist. 6. **Analyze the behavior near $x=1$:** - Numerator near $x=1$ is close to 5 (positive). - Denominator near $x=1$ is $(x-5)(x-1)$, which approaches 0. For $x \to 1^-$, $(x-1)$ is negative, so denominator is negative times negative = positive. For $x \to 1^+$, $(x-1)$ is positive, so denominator is negative times positive = negative. 7. **Conclusion:** - As $x \to 1^-$, the expression tends to $+\infty$. - As $x \to 1^+$, the expression tends to $-\infty$. Therefore, the limit does not exist because the left and right limits are not equal.