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. $$ \lim _{x \rightarrow 0} f(x), \text { for } f(x)=\left\{\begin{array}{l} 2 x+1, \text { if } x \leq 0 \\ 2 x-1, \text { if } x>0 \end{array}\right. $$

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.
$$
\lim _{x \rightarrow 0} f(x), \text { for } f(x)=\left\{\begin{array}{l}
2 x+1, \text { if } x \leq 0 \\
2 x-1, \text { if } x>0
\end{array}\right.
$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Limit

A limit represents the value that a function approaches as its input gets arbitrarily close to a particular point. It provides a way to understand the behavior of a function even at points where the function may not be directly defined.

One-Sided Limits

One-sided limits consider the behavior of a function approaching a specific point from only one direction—either from the left or the right. This is crucial when dealing with functions defined by different rules on either side of the point, as each side may yield a different limit.

Piecewise Functions

Piecewise functions are defined by separate expressions over different intervals of the domain. Understanding how to evaluate and graph these functions is essential since they can exhibit different behaviors, such as varying slopes, intercepts, or discontinuities at the boundaries between pieces.

Jump Discontinuity

A jump discontinuity occurs when the left-hand and right-hand limits exist as separate finite numbers but do not match. This discontinuity highlights that even though each one-sided limit exists, the overall limit at that point does not exist.

Transcript

Please give Ace some feedback