Exercise · ZW-SEQUENTIAL-CONTINUITY-005 · ★☆☆ Basic

Let \(f(x)=\dfrac{x^2-1}{x-1}\) if \(x\ne1\) and \(f(1)=3\). With \(x_n=1+1/n\), what does \(f(x_n)\) converge to?

Choose one option.

Show answer

D. \(2\)

Explanation

For \(x\ne1\), \(f(x)=x+1\), so \(f(x_n)=2+1/n\to2\ne f(1)=3\): there is a discontinuity at \(1\).

Review the concept: Sequential continuity

A function \(f\) is sequentially continuous at \(c\) if, for every sequence \((x_n)\) in the domain converging to \(c\), the sequence of images \(f(x_n)\) converges to \(f(c)\).

\[x_n\to c\ \Longrightarrow\ f(x_n)\to f(c)\]

More practice problems: sequential continuity

ZW-SEQUENTIAL-CONTINUITY-001 Let f(x)=x^2 and x_n=1+1/n. What does f(x_n) converge to?
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ZW-SEQUENTIAL-CONTINUITY-003 Let f be the Dirichlet function (1 at rationals, 0 at irrationals) and x_n=√2/n. What is lim f(x_n)?
ZW-SEQUENTIAL-CONTINUITY-004 If f is continuous at c, what holds?
ZW-SEQUENTIAL-CONTINUITY-006 To prove with the sequential criterion that f is not continuous at c, it suffices to exhibit:
ZW-SEQUENTIAL-CONTINUITY-007 Let f(x)=sin (1/x) for xne0 and f(0)=0. With x_n=(1)/(π/2+2π n), what does f(x_n) converge to?
ZW-SEQUENTIAL-CONTINUITY-008 The sequential criterion says that, for a real function f at a point c of its domain:
ZW-SEQUENTIAL-CONTINUITY-009 Let f(x)=x if x is rational and f(x)=0 if irrational. At which points is it continuous?