Use the graph of f in the figure to find the following...

Use the graph of $f$ in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why.
a. $f(1)$
b. $\lim _{x \rightarrow 1} f(x)$
c. $\lim _{t \rightarrow 1^{-}} f(x)$
d. $\lim _{x \rightarrow 1} f(x)$

Key Concepts

Function Evaluation at a Point

This concept refers to determining the value of a function at a specific input. It involves simply substituting the input into the function and reading its output based on the graph. Understanding this process is essential for distinguishing between the function's defined output at a point and its behavior in the immediate neighborhood of that point.

Limits of a Function

The limit of a function as x approaches a particular value describes the behavior of the function near that point, rather than the function's actual value at the point. This concept helps in understanding situations where the function may not be defined, or its value is different from the expected trend, particularly near points of discontinuity.

One-Sided Limits

One-sided limits examine the behavior of a function as x approaches a certain value either from the left (values smaller than the target) or from the right (values greater than the target). These limits are crucial when dealing with functions that exhibit different behavior on either side of a point, helping to identify the nature of discontinuities such as jumps or gaps.

Discontinuities

Discontinuities occur when a function's behavior is not smooth at a certain point, possibly due to an abrupt jump, hole, or vertical asymptote. Recognizing discontinuities is important for analyzing limits since the existence and values of limits depend on the behavior of the function near the point in question, even if the function's value at that point is defined differently.

Transcript

Please give Ace some feedback