i was just wondering if there was a difference between infinty and indetrminant. my freind said there the same thing. But i thought that for example if you are given a limit for a function, by direct subsitution you find that answer is infity. the conclusion would be that the limit does not exist right? However, if you were given another limit problem and you direct subsitute and find a indetrminent form (0/0)then you choose other techniques ( eg. rationalizing)to solve it. so how could they be the same thing if they are approched differnt ways?
That's a great question! Infinity and indeterminate are related concepts, but they do have distinct meanings in mathematics. Let me explain:
First, let's start with infinity. In mathematics, infinity (∞) represents a concept of limitless or unbounded quantities. It is not a specific number but rather a concept used to describe values that are infinitely large. For example, if you have a function and its output values grow without bounds as the input approaches a certain value, we say that the limit of the function as the input approaches that value is infinity.
On the other hand, indeterminate refers to a specific type of limit that cannot be determined or evaluated directly by simple substitution. It arises when you encounter an expression that does not give a definite value when you plug in a certain input. The most common indeterminate form is 0/0, where the numerator and the denominator both tend to zero.
Now, the reason why these terms can sometimes be confused is that both infinity and indeterminate forms can be encountered in limit problems. However, they are not the same thing. When you evaluate a limit and the result is infinity, it means that the function grows without bounds. This implies that the limit does exist, albeit infinity is considered to be a special kind of value.
On the other hand, when you get an indeterminate form such as 0/0, it means that you cannot determine the value of the limit just by direct substitution. However, this does not mean that the limit does not exist. It could still exist, and you need to use other techniques, such as factoring, rationalizing, or applying L'Hôpital's rule, to determine the actual value of the limit.
So, in summary, infinity represents an unbounded or limitless quantity, while indeterminate forms are specific cases where the value of a limit cannot be determined directly. They are related but not the same thing, and they require different approaches for their evaluation.