Limits typically fail to exist for one of four reasons: The one-sided limits are not equal. The function doesn't approach a finite value (see Basic Definition of Limit). The function doesn't approach a particular value (oscillation). The x - value is approaching the endpoint of a closed interval.

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The derivative measures the steepness of the graph of a function at some particular point on the graph. Thus, the derivative is a slope. (That means that it is a ratio of change in the value of the function to change in the independent variable.)

Limit and Deviation

The concepts of limit and deviation are commonly used in various fields, including mathematics, statistics, and science. They help to understand the behavior and variability of data, functions, or processes. Let's delve into the details of each:

Limit:
- In mathematics, the limit is a fundamental concept used to describe the behavior of a function or sequence as its input or index approaches a particular value.
- The limit of a function at a specific point is defined as the value that the function approaches as the input approaches that point.
- It represents the value that a function "tends to" or "approaches" as the input gets arbitrarily close to a given value.
- The limit can exist and be finite, infinite, or undefined.
- Mathematically, the limit of a function f(x) as x approaches a certain value (say 'a') is denoted as lim(x → a) f(x) or simply lim f(x).

Deviation:
- Deviation, in statistics, refers to the difference between a data point and a reference point (such as the mean or median) within a dataset.
- It measures the spread or dispersion of the data values around the central tendency.
- Deviation can be calculated for individual data points or for a group of data points.
- The deviation from the mean is commonly used to assess how much a data point differs from the average value.
- Positive deviations indicate values above the reference point, while negative deviations indicate values below it.

Key Points:
- Limit is a mathematical concept that describes the behavior of a function or sequence as the input approaches a specific value.
- Deviation is a statistical measure that quantifies the difference between a data point and a reference point within a dataset.
- Limit can be finite, infinite, or undefined, while deviation can be positive or negative depending on whether it is above or below the reference point.
- Limit is denoted as lim(x → a) f(x), while deviation is calculated by subtracting the reference point from the data point.
- Both concepts play crucial roles in understanding the properties and variability of functions and datasets.