B.E. SEM 1 ELECTRICAL DIPARTMENT DIVISION K GROUP(41-52) B.E. SEM 1 ELECTRICAL DIPARTMENT DIVISION K GROUP(41-52)
Zero divided by zero can not be evaluated. The limit may or may not exist, and is called an indeterminate form. Consider: or If we try to evaluate by direct substitution, we get: In the case of the first limit, we can evaluate it by factoring and canceling: This method does not work in the case of the second limit. Indeterminate forms
I NDETERMINATE F ORMS The expressions of the form which all are undefined and are called Indeterminate forms.
L’H ÔPITAL ’ S R ULE
Example: If it’s no longer indeterminate, then STOP differentiating! If we try to continue with L’Hôpital’s rule: which is wrong!
I NDETERMINATE FORM Evaluate
This approaches We already know that but if we want to use L’Hôpital’s rule: INDETERMINATE FORM Rewrite as a ratio!
If we find a common denominator and subtract, we get: Now it is in the form This is indeterminate form L’Hôpital’s rule applied once. Fractions cleared. Still INDETERMINATE FORM Rewrite as a ratio! L’Hôpital again. Answer:
Indeterminate Forms: Evaluating these forms requires a mathematical trick to change the expression into a ratio. We can then write the expression as a ratio, which allows us to use L’Hôpital’s rule. We can take the log of the function as long as we exponentiate at the same time. Then move the limit notation outside of the log. Indeterminate Powers
I NDETERMINATE F ORM 0 0
I NDETERMINATE F ORM ∞ 0
I NDETERMINATE F ORM 1 ∞
IMPROPAR INTIGRAL TYPE-1 INFINTE INTERVALS TYPE-2 DISCONTINUOS INTEGRANDS
DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1
DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2
DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2
A COMPARISON TEST FOR IMPROPER INTEGRALS Sometimes it is impossible to find the exact value of an improper integral and yet it is important to know whether it is convergent or divergent. In such cases the following theorem is useful. Although we state it for Type 1 integrals, a similar theorem is true for Type 2 integrals.