It depends on how you define "evaluated." If, when you attempt to evaluate a limit, you get an indeterminate form, there are techniques that you can use to either a) evaluate the limit (get a number), or b) say that the limit doesn't exist (which includes ##\infty## as the "value" of the limit).darkchild said:Is it true that every limit that takes on an indeterminate form can be evaluated?
We don't say that a limit problem "has a solution." Equations and inequalities have solutions. A limit can be a) a finite number, b) unbounded, or c) not exist at all.darkchild said:Is it proper to say that a limit problem has a solution if the limit does not exist?
Mark44 said:It depends on how you define "evaluated." If, when you attempt to evaluate a limit, you get an indeterminate form, there are techniques that you can use to either a) evaluate the limit (get a number), or b) say that the limit doesn't exist (which includes ##\infty## as the "value" of the limit).
Like I said, when you get an indeterminate form, there are techniques (such as L'Hopital's Rule or algebraic techniques) that you can use to evaluate the limit or say that it doesn't exist. I wouldn't call this "simplifying" the limit expression, though. L'Hopital's Rule applies only to the ##[\frac{-\infty}{\infty}]## and ##[\frac 0 0]## indeterminate forms. Other indeterminate forms, such as ##[1^{\infty}]##, require different techniques.darkchild said:Ok, then is it appropriate to say that every indeterminate form can be simplified?
Limits and indeterminate forms are mathematical concepts used to describe the behavior of a function as its inputs approach a certain value. A limit represents the value that a function "approaches" as its inputs get closer and closer to a particular value. Indeterminate forms are expressions that cannot be evaluated using traditional algebraic methods because they result in undefined or infinite values.
The three main types of limits are one-sided limits, two-sided limits, and infinite limits. One-sided limits describe the behavior of a function as its inputs approach a particular value from either the positive or negative direction. Two-sided limits describe the behavior of a function as its inputs approach a particular value from both the positive and negative directions. Infinite limits describe the behavior of a function as its inputs approach infinity or negative infinity.
Limits can be evaluated using various techniques, including direct substitution, algebraic manipulation, and using theorems such as the Squeeze Theorem or the Intermediate Value Theorem. In some cases, limits may require more advanced techniques such as L'Hopital's Rule or series expansions.
Some common indeterminate forms include 0/0, ∞/∞, 0*∞, and ∞-∞. These forms arise when evaluating limits that involve functions with asymptotes, discontinuities, or undefined values.
Indeterminate forms are important because they allow us to evaluate limits of functions that would otherwise be undefined or infinite. They also help us understand the behavior of functions near certain points and can provide insights into the nature of a function's graph. Indeterminate forms are also essential in many areas of mathematics, including calculus, differential equations, and complex analysis.