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Indeterminate Form - Zero Times Infinity

__Category__: Differential Calculus, Trigonometry
"Published in Newark, California, USA"
Evaluate the limit for

__Solution__:

Consider the given equation

Substitute the value of x to the above equation, we have

Since the answer is ∞∙0 which is also another type of Indeterminate Form, it is not accepted in Mathematics as a final answer. We know that any number multiply by zero is always equal to zero but there's an exception, which is infinity. Therefore, we cannot say that infinity times zero is zero. In this type of Indeterminate Form, you cannot use the L'Hopital's Rule because the L'Hopital's Rule is applicable for the Indeterminate Forms like 0/0 and ∞/∞. In this case, let's rewrite the given equation as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form at this time is ∞/∞, then we can apply the L'Hopital's Rule to the above equation as follows

Since the trigonometric functions at the above equation are all cancelled and the constant is left, then we cannot substitute the value of x. Therefore,