Mathematics

The key here is understanding the difference between a function's value and its limit. Even though $h(1) = 0$ by definition, what matters for continuity is whether that value matches what the function approaches as $x$ gets arbitrarily close to 1. Since $\frac{1}{x-1}$ shoots toward $\pm\infty$ near $x=1$, the limit doesn't exist—so $h$ has a discontinuity there despite being defined. The function is removable only if we could redefine it at one point to make it continuous; here, no single value works.

The key here is understanding the difference between a function's value and its limit. Even though $h(1) = 0$ by definition, what matters for continuity is whether that value matches what the function approaches as $x$ gets arbitrarily close to 1. Since $\frac{1}{x-1}$ shoots toward $\pm\infty$ near $x=1$, the limit doesn't exist—so $h$ has a discontinuity there despite being defined. The function is removable only if we could redefine it at one point to make it continuous; here, no single value works.