Exercise · ZW-COMPLEX-LIMIT-009 · ★★★ Advanced
If \(|f(z)-(2+i)|\le 3|z|\) for all \(z\), what follows about \(\lim_{z\to 0}f(z)\)?
Choose one option.
Show answer
- C. it equals \(2+i\)
Explanation
Since \(3|z|\to 0\), the modulus \(|f(z)-(2+i)|\) tends to \(0\) (squeeze theorem), i.e. \(f(z)\to 2+i\).
Review the concept: Complex limit
The limit \(L\) of a function \(f\) at \(z_0\) is the complex number that \(f(z)\) gets as close to as desired (\(\varepsilon\)) when \(z\) comes within distance \(\delta\) of \(z_0\), from any direction.
\[\lim_{z\to z_0}f(z)=L\iff\forall\varepsilon>0\ \exists\delta>0:\ 0<|z-z_0|<\delta\Rightarrow|f(z)-L|<\varepsilon\]
More practice problems: complex limit
- ZW-COMPLEX-LIMIT-001 Compute lim _z→ 1+i(z^2+1).
- ZW-COMPLEX-LIMIT-002 Compute lim _z→ 2i(z^2+4)/(z-2i).
- ZW-COMPLEX-LIMIT-003 Compute lim _z→ i(z^4-1)/(z-i).
- ZW-COMPLEX-LIMIT-004 Compute lim _z→ 0(z^2)/(|z|).
- ZW-COMPLEX-LIMIT-005 If lim _z→ z_0f(z)=1+i and lim _z→ z_0g(z)=2-i, what is lim _z→ z_0f(z)g(z)?
- ZW-COMPLEX-LIMIT-006 About lim _z→ 0(bar z)/(z) it is claimed that:
- ZW-COMPLEX-LIMIT-007 Let f(z)=(x^2-y^2)/(x^2+y^2) with z=x+iy. Which is correct about lim _z→ 0f(z)?
- ZW-COMPLEX-LIMIT-008 The value f(z_0) of a function at the point where the limit is taken: