Use a graph of f to estimate $\underset{x\to a}{\lim }f\left(x\right)$ or to show that the limit does not exist. Evaluate f(x) near $x=a$ to support your conjecture.
$f\left(x\right)=\frac{1-\cos (2x-2)}{{(x-1)}^2};a=1$

Key Concepts

Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a certain value. It is denoted as lim(x→a) f(x) and indicates what value f(x) approaches as x gets closer to a. Understanding limits is crucial for analyzing continuity and differentiability, as well as for evaluating functions that may not be defined at certain points.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For the function f(x) to be continuous at x = a, it must satisfy three conditions: f(a) must be defined, the limit as x approaches a must exist, and both must be equal. Discontinuities can lead to limits that do not exist or are undefined.

Graphical Interpretation of Limits

Using a graph to estimate limits involves observing the behavior of the function as it approaches a specific x-value. By analyzing the graph, one can identify trends, such as whether the function approaches a finite value, diverges, or oscillates. This visual approach aids in understanding the concept of limits and can provide insights into the existence or non-existence of limits at certain points.