Finding Limits Algebraically (cont.)
Not all limits can be algebraically derived from the function through substitution or factoring. There are also "special cases" that have separate methods to be used, such as trigonometric limits, limits of absolute value functions (if there is an "x" in the denominator), limits of functions with radicals in the numerator, and limits of complex fractions.
Here, we find that there is a 0 in the denominator AND that the function can not be factored. Thus, we can conclude that the limit does not exist. If graphed, the function exposes a "jump" in the line.
Radicals in the Numerator:
Limits can not be found while there is a root function in the numerator. To remove the root, we must rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator.
Here, we find that, when x is plugged in, there is a 0 in the denominator. To remove the radical, we must multiply the numerator and denominator by the conjugate, which in this case would be √x + 2
After multiplying, we see that there is (x-4) in both the numerator and denominator, meaning we can cross out both of them while leaving a 1 in the numerator. Lastly, plug in and solve for 4 and you are left with the limit: 1/4
To solve trigonometric functions such as the one below, we must use trigonometric substitution to re-write the function so that plugging x in does not result in "0" in the denominator.
First and foremost, we know that tan is (sin/cos). We could:
1) Substitute "sin/cos" in place of "tan(x)," leaving a fraction in the numerator. After we move cos to the denominator, we are left with "sin(x)/xcos(x)"
2)Separate the function into two parts: sin(x)/x and 1/cos(x).
3) Evaluate the function: we know that the limit of sin(x)/x as x approaches 0 is equal to 1. We also know that the limit of 1/cos(x) as x approaches 0 is 1. Lastly, multiply the two together and you are left with the limit of tan(x)/x: 1.
When dealing with a fraction within a fraction, there is a method of approaching it: keep, change, flip. In practice, you would find the common denominator of the fractions within the numerator, change the division sign to multiplication, flip the fractions in the denominator, and multiply.