Question

Find the value of $\lim_{x\to 3} \frac{x-2}{x-3}$.

Answer 1

The limit does not exist.

Answer 2

The limit $\lim_{x\to 3} \frac{x-2}{x-3}$ results in the indeterminate form $\frac{1}{0}$ upon direct substitution. Evaluating the left-hand limit (LHL) gives $\lim_{x\to 3^-} \frac{x-2}{x-3} = -\infty$, because the numerator approaches $1$ and the denominator approaches $0$ from the negative side. Evaluating the right-hand limit (RHL) gives $\lim_{x\to 3^+} \frac{x-2}{x-3} = +\infty$, because the numerator approaches $1$ and the denominator approaches $0$ from the positive side. Since LHL $\neq$ RHL, the limit does not exist.