key term - $rac{ ext{infty}}{ ext{infty}}$

Definition

$rac{ ext{infty}}{ ext{infty}}$ is an indeterminate form that arises when the numerator and denominator of a fraction both approach positive or negative infinity. This term is particularly relevant in the context of finding limits, as the properties of limits can be used to evaluate expressions in this form.

5 Must Know Facts For Your Next Test

1. The expression $rac{ ext{infty}}{ ext{infty}}$ is considered an indeterminate form because it does not have a definite numerical value.
2. When evaluating limits involving $rac{ ext{infty}}{ ext{infty}}$, the properties of limits must be applied to determine the actual limit value.
3. Expressions in the form $rac{ ext{infty}}{ ext{infty}}$ can arise when a function's numerator and denominator both approach positive or negative infinity as the input variable approaches a particular value.
4. Applying L'Hôpital's rule is a common technique used to evaluate limits in the form $rac{ ext{infty}}{ ext{infty}}$.
5. Understanding the behavior of functions near points where $rac{ ext{infty}}{ ext{infty}}$ occurs is crucial for determining the continuity of the function at those points.

Review Questions

• Explain why the expression $rac{ ext{infty}}{ ext{infty}}$ is considered an indeterminate form.
  • The expression $rac{ ext{infty}}{ ext{infty}}$ is considered an indeterminate form because it does not have a definite numerical value. When the numerator and denominator of a fraction both approach positive or negative infinity, the actual limit value cannot be determined by simply dividing the two. Instead, the properties of limits must be applied to evaluate the expression and determine the actual limit value.
• Describe how the properties of limits can be used to evaluate expressions in the form $rac{ ext{infty}}{ ext{infty}}$.
  • When evaluating limits involving the expression $rac{ ext{infty}}{ ext{infty}}$, the properties of limits can be applied to simplify the expression and determine the actual limit value. This may involve techniques such as applying L'Hôpital's rule, which states that if the limit of a fraction with both the numerator and denominator approaching infinity is in the form $rac{ ext{infty}}{ ext{infty}}$, then the limit is equal to the limit of the derivative of the numerator divided by the derivative of the denominator. By using these limit properties, the indeterminate form $rac{ ext{infty}}{ ext{infty}}$ can be resolved, and the true limit value can be found.
• Analyze the relationship between the expression $rac{ ext{infty}}{ ext{infty}}$ and the continuity of a function.
  • The expression $rac{ ext{infty}}{ ext{infty}}$ is closely related to the continuity of a function. When a function's numerator and denominator both approach positive or negative infinity at a particular point, the function may exhibit a discontinuity at that point. Understanding the behavior of the function near these points where $rac{ ext{infty}}{ ext{infty}}$ occurs is crucial for determining the continuity of the function. By applying the properties of limits and evaluating the expression $rac{ ext{infty}}{ ext{infty}}$, one can assess whether the function is continuous or discontinuous at the point in question, which is an essential skill in the study of limits and continuity.

Related terms

Indeterminate Form:

An expression that cannot be immediately evaluated to a definite value, such as $rac{0}{0}$ or $rac{ ext{infty}}{ ext{infty}}$.

Limit Properties: A set of rules and theorems that can be used to evaluate the limit of an expression, including the ability to handle indeterminate forms.

Continuity:

A property of a function where the function's value at a point is equal to the limit of the function as it approaches that point from both sides.