Compute the limit (exponents with variable)

In summary, the conversation revolves around finding the limits of a given sequence and the methods used to solve it. The individual limits are found to be +∞ and when divided, the result is an indeterminant value. The conversation also suggests trying a different approach using exponents.
  • #1

Homework Statement


Compute the limits of the following sequence (show your work):

3n+17n+3
4n5n-2

(sorry if it looks funny ... the top line is divided by the bottom line)

The Attempt at a Solution


I was thinking that I should get the exponents the same and add/subtract the exponents, but I don't remember how to, or if this is even possible.
 
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  • #2
I think I figured out a bit more. I have learned that I can split up the limit, so it will look like:
(lim3n+1)(lim7n+3) ÷ (lim4n)(lim5n-2)

and each of these individual limits equals +∞.

So I have +∞ ÷ +∞. Does this still equal infinity, 0, or does not exist?
 
  • #3
PirateFan308 said:
I think I figured out a bit more. I have learned that I can split up the limit, so it will look like:
(lim3n+1)(lim7n+3) ÷ (lim4n)(lim5n-2)

and each of these individual limits equals +∞.

So I have +∞ ÷ +∞. Does this still equal infinity, 0, or does not exist?

Try this. 2^n/3^n=(2/3)^n. What's the limit of that? How about 3^n/2^n? There's a reason why infinity/infinity is called 'indeterminant'.
 

1. What is a limit in terms of exponents with variables?

A limit in terms of exponents with variables is a mathematical concept that describes the value that a function approaches as the input variable gets closer and closer to a specific value. In other words, it is the value that a function "approaches" but may not necessarily reach when the input variable gets infinitely close to a specific value.

2. How do you compute a limit with exponents and variables?

To compute a limit with exponents and variables, you can use algebraic techniques such as factoring, simplifying, or combining like terms. You can also use graphical methods or numerical methods such as plugging in values to a table or calculator. In some cases, you may need to use more advanced techniques such as L'Hopital's rule or the squeeze theorem.

3. What are the common types of limits involving exponents and variables?

The common types of limits involving exponents and variables include polynomial limits, rational limits, radical limits, logarithmic limits, and exponential limits. These types of limits may involve different techniques for computing them, but the underlying concept remains the same.

4. What are the properties of limits with exponents and variables?

The properties of limits with exponents and variables include the sum, difference, product, and quotient properties, as well as the power rule and the constant multiple rule. These properties allow us to simplify and evaluate more complex limits by breaking them down into smaller, more manageable parts.

5. Why are limits with exponents and variables important in mathematics?

Limits with exponents and variables are important in mathematics because they help us understand the behavior of functions and their graphs. They allow us to determine if a function is continuous at a certain point or if it has a horizontal or vertical asymptote. Limits are also crucial in calculus, as they are used to define derivatives and integrals.

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