Test for pointwise and uniform convergence
Test for pointwise and uniform convergence of $u_n(x)=\sqrt{x^2+\dfrac1{n^2}}$ on $(-\infty,\infty)$.
Explanation of Method
Since $1/n^2\to0$, the pointwise limit is $u(x)=|x|$ — notably not differentiable at $0$, yet still a legitimate uniform limit. Rationalizing the difference $u_n(x)-|x|$ turns it into a fraction that is maximized exactly at $x=0$ (where the "correction" term $1/n^2$ contributes the most, relatively speaking), giving an explicit bound $1/n\to0$ independent of $x$.
- Find the pointwise limit $u(x)=|x|$.
- Rationalize the difference $u_n(x)-|x|$ to isolate the error term.
- Maximize the error over all of $\mathbb R$ and confirm it tends to $0$.
Step 1 · Pointwise limit
For fixed $x$: $\sqrt{x^2+1/n^2}\to\sqrt{x^2}=|x|$ as $n\to\infty$.
Step 2 · Rationalize the error
Since $\sqrt{x^2+1/n^2}>|x|$ always: $$ \sqrt{x^2+\tfrac1{n^2}}-|x| = \frac{\left(x^2+\frac1{n^2}\right)-x^2}{\sqrt{x^2+\frac1{n^2}}+|x|} = \frac{1/n^2}{\sqrt{x^2+1/n^2}+|x|}. $$
Step 3 · Maximize over $\mathbb R$
The denominator is minimized (making the fraction largest) when $|x|$ is smallest, i.e. at $x=0$: $$ \sup_{x\in\mathbb R}\left|u_n(x)-|x|\right| = \frac{1/n^2}{\sqrt{1/n^2}+0} = \frac{1/n^2}{1/n} = \frac1n \to 0. $$ Uniform convergence holds on all of $\mathbb R$.
Summary Table
| Quantity | Value |
|---|---|
| Pointwise limit | $u(x)=|x|$ |
| Location of max error | $x=0$ |
| $\sup_{\mathbb R}|u_n-u|$ | $1/n\to0$ |
Additional Notes
- This is a striking example where uniform convergence holds on the entire unbounded real line, not merely on a compact sub-interval — a stronger conclusion than in Problems 176 or 177, made possible because the error term $1/n^2$ shrinks the "smoothing" uniformly regardless of how large $|x|$ is.
- Even though the limit function $u(x)=|x|$ fails to be differentiable at $x=0$ while every $u_n$ is smooth there, uniform convergence is unaffected — smoothness, unlike continuity, is not automatically inherited from uniform limits.
- Each $u_n(x)=\sqrt{x^2+1/n^2}$ is itself a classical smooth approximation to the absolute value function, widely used in optimization and numerical analysis precisely because of this uniform (and, as shown, quantifiably fast) convergence.
Result
Converges uniformly to $u(x)=|x|$ on all of $\mathbb R$: