Use the graph to determine each limit (if it exists). Then...

Use the graph to determine each limit (if it exists). Then identify another function that agrees with the given function at all but one point.
$g(x)=\frac{-2 x^{2}+x}{x}$
(a) $\lim _{x \rightarrow 0} g(x)$
(b) $\lim _{x \rightarrow-1} g(x)$
(c) $\lim _{x \rightarrow-2} g(x)$

Key Concepts

Limit of a Function

This principle in calculus measures the behavior of a function as its input approaches a particular point, independent of the function’s behavior exactly at that point. It is fundamental for understanding continuity, and establishing whether a function approaches a unique value near a point, even if the function may be undefined or discontinuous at that exact point.

Removable Discontinuity

A removable discontinuity occurs when a function is not defined at a specific point, yet the limit exists as the variable approaches that point. Often, it arises from factors in the numerator and denominator that cancel out, making it possible to redefine the function to 'fill in' the missing value, thereby restoring continuity.

Simplification of Rational Expressions

Simplifying rational expressions involves canceling common factors in the numerator and denominator. It is a key technique when evaluating limits, particularly when the direct substitution yields an indeterminate form, as it can reveal the actual behavior of the function near the point of interest.

Graphical Interpretation of Limits

Using a graph to evaluate limits involves analyzing how the function behaves as its input gets arbitrarily close to a target value. This visual approach helps determine the existence and value of limits, identify discontinuities, and observe overall trends and asymptotic behavior of the function.

Transcript

Please give Ace some feedback