Definition. Let \( f(x)\) be defined for all \( x \ne c\) over an open interval \( (a,b)\) containing \( c \). Let \( L \) be a real number. Then
\[ \lim_{x \to c} f(x) = L \] means that, for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that, whenever \( c-\delta < x < c \) or \(c < x < c+ \delta \), one has \( L-\varepsilon < f(x) < L+\varepsilon \).

Intuitively, \( \varepsilon \) controls how far we “zoom in” in the \( y \)-direction while \( \delta \) controls how far we “zoom in” in the \( x \)-direction. So the definition above says that, no matter how far we zoom in in the \( y \)-direction, we can zoom in in the \( x \)-direction far enough so that the function is contained entirely in the box (except possibly at \( x = c\)). Graphically, this is represented by the function “exiting” the box only through the sides (and not the top or bottom).

Formally, when checking this definition, we want to answer the following question: Is it possible to select \( \delta \) in terms of \( \varepsilon \) so that the values \( f(x) \) will always be between \( L-\varepsilon \) and \(L+\varepsilon\) when the input \(x\) falls within the zooming window—that is, when \( c-\delta < x < c+\delta \) (except possibly at \( x = c \))? Is this possible for all \( \varepsilon > 0 \)?

Use the SageMath app below to complete the lab report.

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