Removable discontinuities

Task number: 2987

Determine whether the following functions can be made be continuous by defining a value at \(x=0\). The functions are defined on \(\mathbb R\setminus\{0\}\) by the following formulas:

Variant 1
\( f(x)=e^{|x|} \)
Resolution
\( \displaystyle \lim_{x\to 0^+}e^{|x|}= \lim_{x\to 0^+}e^{x}=1 \) and also \( \displaystyle \lim_{x\to 0^-}e^{|x|}= \lim_{x\to 0^-}e^{-x}= \lim_{x\to 0^+}e^{x}=1 \), so the function \(f\) can be made continuous by defining \(f(0)=1\).
Result
We can define \(f(0)=1\).
Variant 2
\(f(x)=\arctan\frac1{x^2}\)
Variant 3
\( f(x)=\frac{1-\cos x}{x^3} \)
Variant 4
\( f(x)=x^3\sin^2\frac1x \)

Difficulty level: Easy task (using definitions and simple reasoning)

Removable discontinuities

Task number: 2987

Variant 1

Resolution

Result

Variant 2

Variant 3

Variant 4