how do you solve limits that have infinities!?
You have to be much more specific in what you need help with.
bob i need help paraphrasing
To solve limits that involve infinities, you need to determine the behavior of the function as the input approaches infinity or negative infinity. Here are two common cases:
1. Infinite Limits: Consider a limit where the function grows without bound as the input approaches a particular value (e.g., as x approaches infinity). To solve this, you can use the following techniques:
a. Determine the highest power of the variable in the numerator and denominator of the function. If the power in the numerator is higher, then the limit is positive or negative infinity, depending on the sign of the leading coefficient. If the power in the denominator is higher, then the limit is zero.
b. For rational functions (where numerator and denominator are polynomials), compare the degrees of the numerator and denominator. If the degrees are the same, divide both coefficients by the highest power of the variable and apply the above rule.
c. Some functions may have more complex behavior as the input approaches infinity, such as oscillations or different rates of growth. In such cases, you might need to apply additional techniques like L'Hôpital's rule or algebraic manipulation to simplify the function and find the limit.
2. Limits with Infinity in Algebraic Expressions: These limits typically involve indeterminate forms like ∞/∞, 0/0, or ∞ - ∞. To solve these limits, you can use the following methods:
a. Simplify the expression by factoring, canceling common factors, or applying algebraic properties. This may help transform the expression into a form that is easier to evaluate.
b. If manipulation doesn't lead to an immediate solution, consider applying L'Hôpital's rule. This rule allows you to differentiate the numerator and denominator independently and take the limit again. Repeat this process until you reach a solvable form.
c. In certain cases, it may be helpful to use special trigonometric or logarithmic identities that can simplify the expression and facilitate evaluation.
Remember that these methods are not exhaustive, and each limit problem may require different techniques. Practice and familiarity with different types of limits will sharpen your problem-solving skills.