Sketch graphs by hand and use them to make approximations...

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow-2} \frac{x^{2}+x-2}{x+2}
$$

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Key Concepts

Limit of a Function

The limit of a function as the variable approaches a particular value describes the value the function gets arbitrarily close to, regardless of whether the function is defined at that point. It is a foundational concept in calculus, underlying the definitions of continuity, derivatives, and integrals.

One-Sided Limits

One-sided limits analyze the behavior of a function as the variable approaches a specified point from one side only—either from the left or the right. They are crucial when a function exhibits different behavior on either side of a point, allowing us to describe and understand discontinuities or directional trends.

Removable Discontinuity

A removable discontinuity occurs when a function is not defined at a point, yet the limit exists because the apparent discontinuity can be 'removed' by simplifying the function's expression. This often happens when factors cancel out algebraically, indicating that the discontinuity is a point-wise issue rather than a fundamental break in the function's behavior.

Graphical Analysis of Limits

Graphical analysis involves using hand-drawn or plotted graphs to visually assess the behavior of a function near a point of interest. This approach aids in approximating limits, identifying trends, and recognizing features such as discontinuities or asymptotic behavior.

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