Limit as x goes to infinity (algebraic)

In summary, the limit as x approaches infinity of (x+2)/(sqrt(81x^2+15)) can be simplified by dividing both the numerator and denominator by x, leading to a final answer of 1/9. This method can be applied to other rational polynomial limits as well by dividing by the highest factor.
  • #1
mbisCool
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Homework Statement


limx->infinity (x+2)/(sqrt(81x^2+15))


Homework Equations





The Attempt at a Solution


The only thing i could think of doing was rationalizing the denominator to get (x+2)sqrt(81x^2+15) / 81X^2+15 however I am pretty sure this is the wrong route cause there doesn't seem to be anywhere to go from here.

Any help would be greatly apprciated :D the only problems i seem to ever have difficulty with are with roots in the numerator or denominator. any info in that regard would be of use as well.
 
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  • #2
divide top and bottom by x and write an x inside the root
 
  • #3
what exactly do you mean by write an x inside the root. if i divide top and bottom by x i get ((x+2)/x) / ((sqrt(81x^2+15))/x). from here I am stuck again
 
  • #4
alright its 1/9
 
  • #5
note that this trick generalizes to all rational polynomial limits to infinity, just divide by the highest factor.
 

FAQ: Limit as x goes to infinity (algebraic)

What is a limit as x goes to infinity?

A limit as x goes to infinity is a mathematical concept that describes the behavior of a function as the input variable (x) approaches infinity. It determines the value that a function approaches as its input variable becomes infinitely large.

How is a limit as x goes to infinity calculated?

The limit as x goes to infinity can be calculated using the following steps:

  1. Simplify the algebraic expression by canceling out any common factors.
  2. Substitute infinity (represented by the symbol ∞) for x in the simplified expression.
  3. Solve the resulting expression to determine the limit as x goes to infinity.

What is the difference between a limit as x goes to infinity and a limit at infinity?

A limit as x goes to infinity is a specific type of limit that describes the behavior of a function as the input variable approaches infinity. A limit at infinity, on the other hand, is a limit that is calculated when the input variable is already at infinity.

What types of functions have limits as x goes to infinity?

Most functions have limits as x goes to infinity, including polynomial functions, rational functions, and exponential functions. However, some functions, such as oscillating functions, do not have a limit as x goes to infinity.

How does the graph of a function relate to its limit as x goes to infinity?

The limit as x goes to infinity can be visualized by examining the behavior of the function's graph as the input variable approaches infinity. If the function's graph approaches a specific value or asymptote as x increases, then that value is the limit as x goes to infinity. If the graph has no specific value or asymptote, then the limit as x goes to infinity does not exist.

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