Understanding continuity in terms of limits - AP Calculus AB

The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

does not exist, because .

For a function to be continuous the following criteria must be met:

1. The function must exist at the point (no division by zero, asymptotic behavior, negative logs, or negative radicals).
2. The limit must exist.
3. The point must equal the limit. (Symbolically, ).

It is easiest to first find any points where the function is undefined. Since our function involves a fraction and a natural log, we must find all points in the domain such that the natural log is less than or equal to zero, or points where the denominator is equal to zero.

To find the values that cause the natural log to be negative we set

Therefore, those x values will yield our points of discontinuity. Normally, we would find values where the natural log is negative; however, for all the function is positive.

Unless we are explicitly told so, via graph, information, or otherwise, we cannot assume is continuous at unless , which is required for to be continuous at .

We cannot assume anything about the existence of , because we do not know what is, or its end behavior.

The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

does not exist, because .

For a function to be continuous the following criteria must be met:

1. The function must exist at the point (no division by zero, asymptotic behavior, negative logs, or negative radicals).
2. The limit must exist.
3. The point must equal the limit. (Symbolically, ).

It is easiest to first find any points where the function is undefined. Since our function involves a fraction and a natural log, we must find all points in the domain such that the natural log is less than or equal to zero, or points where the denominator is equal to zero.

To find the values that cause the natural log to be negative we set

Therefore, those x values will yield our points of discontinuity. Normally, we would find values where the natural log is negative; however, for all the function is positive.