Finding the exact value of a limit

  • MHB
  • Thread starter NavalMonte
  • Start date
  • Tags
    Limit Value
In summary, the conversation discusses finding the limit of arctan(-2x^3+3x-4) as x approaches infinity. The advice given is to first find the limits of -2x^3+3x-4 and arctan(x) separately, then combine the results to find the limit of the entire expression. After factoring the largest factor of x from the polynomial, it is determined that the limit of -2x^3+3x-4 as x approaches infinity is negative infinity. Therefore, the limit of arctan(-2x^3+3x-4) is equal to -pi/2 as x approaches infinity.
  • #1
NavalMonte
7
0
I'm having a hard time starting this problem

lim of arctan(-2x^3+3x-4x) as x approaches infinity

Any help would be appreciated
 
Physics news on Phys.org
  • #2
Start by finding the limits of $-2x^3+3x-4x$ when $x\to\infty$ (do you mean $-2x^3+3x^2-4x$?) and $\arctan(x)$ when $x\to\pm\infty$.
 
  • #3
Evgeny.Makarov said:
Start by finding the limits of $-2x^3+3x-4x$ when $x\to\infty$ (do you mean $-2x^3+3x^2-4x$?) and $\arctan(x)$ when $x\to\pm\infty$.

I'm sorry, it's actually written as:

lim arctan($-2x^3+3x-4$)
x->∞
 
  • #4
The advice is the same: see where the argument of the arctangent tends to and then see what values arctangent takes there.
 
  • #5
Evgeny.Makarov said:
The advice is the same: see where the argument of the arctangent tends to and then see what values arctangent takes there.

I factored the largest factor of x from the polynomial and got:

lim $x^3$=∞
x->∞

and

lim $(-2+\dfrac{3}{x^2}-\dfrac{4}{x^2})$=-2
x->∞

Would that make the:
lim arctan (-2) =lim arctan($-2x^3+3x-4$)
x->∞...x->∞Edit: I just realized that:
lim $-2x^3+3x-4$ = -∞
x->∞

Therefore,
lim arctan($-2x^3+3x-4$)= -$\dfrac{\pi}{2}$
x->∞

Would that be correct or am I totally off base?
 
Last edited:
  • #6
I can only repeat what I have said: find the limit of the arctangent's argument and then find what arctangent is like near that value. You, instead, found the limit of the arctangent's argument divided by $x^3$ and substituted it into arctangent. I don't understand why you divided the argument by $x^3$.

Edit: I just saw your edit, and it is correct. You may also see this discussion on StackExchange for a similar example.
 

FAQ: Finding the exact value of a limit

What is the purpose of finding the exact value of a limit?

The purpose of finding the exact value of a limit is to determine the behavior of a function as the input values approach a specific point. This allows us to make accurate predictions and calculations in various mathematical and scientific fields.

How do you solve for the exact value of a limit?

To solve for the exact value of a limit, we use various techniques such as algebraic manipulation, substitution, and L'Hopital's rule. We also need to use our knowledge of the properties of limits, such as continuity and differentiability, to accurately determine the value.

Can the exact value of a limit always be found?

No, the exact value of a limit cannot always be found. In some cases, the limit may be undefined or may approach infinity, making it impossible to determine an exact value. However, we can still determine the behavior of the function and its limit using other methods, such as graphing or estimation.

What are some common mistakes to avoid when finding the exact value of a limit?

Some common mistakes to avoid when finding the exact value of a limit include forgetting to check for any discontinuities, incorrectly applying L'Hopital's rule, and not simplifying the expression before taking the limit. It is also important to pay attention to the given conditions and restrictions of the function.

How does finding the exact value of a limit relate to real-world applications?

Finding the exact value of a limit has many real-world applications, such as in physics, engineering, and economics. It allows us to make accurate predictions and calculations for various scenarios, such as determining the maximum capacity of a structure or the optimal production level in a business.

Similar threads

Back
Top