Solve Multi Var Limit: Help Appreciated!

In summary: The Attempt at a SolutionHad this on a test today, honestly not sure how to evaluate. I know you can pass the limit to the inside of arctan but I can't see how the inside goes to infinity. Any help would be appreciated I just want to understand how to do it!In summary, the problem asks you to find the limit, but doesn't ask you to prove that this value is the limit. Proving the limit would require the use of a delta-epsilon proof or a theorem such as the squeeze theorem.
  • #1
Scrope
5
0

Homework Statement


afabe40fefbef53a39ed240e35d877cc
https://gyazo.com/268bef206850bfbf30fb0cca3f783599 <----- The question

Homework Equations

The Attempt at a Solution


Had this on a test today, honestly not sure how to evaluate. I know you can pass the limit to the inside of arctan but I can't see how the inside goes to infinity. Any help would be appreciated I just want to understand how to do it!
 
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  • #2
I don't get infinity for the inside, either. What is the answer that they gave? Did they get pi/2 for the limit? What answer did you get for the inside (and the limit)?
 
  • #3
the answer given is pi/2 for the overall answer and +infinity for the inside
 
  • #4
The only advice I have received so far is to take note that the bottom is the square of the distance from (0,1) but I can’t figure out what they mean
 
  • #5
Scrope said:

Homework Statement


afabe40fefbef53a39ed240e35d877cc
https://gyazo.com/268bef206850bfbf30fb0cca3f783599 <----- The question

Homework Equations

The Attempt at a Solution


Had this on a test today, honestly not sure how to evaluate. I know you can pass the limit to the inside of arctan but I can't see how the inside goes to infinity. Any help would be appreciated I just want to understand how to do it!
The numerator inside the brackets is approaching 1, and the denominator is approaching 0 from the positive side. It seems pretty clear to me that the part in brackets is approaching ##+\infty##, so the overall limit is what?
 
  • #6
Mark44 said:
The numerator inside the brackets is approaching 1, and the denominator is approaching 0 from the positive side. It seems pretty clear to me that the part in brackets is approaching ##+\infty##, so the overall limit is what?

the way I had learned it was that the only way to prove a limit existed was through methods such as delta-epsilon definition, switching to R or Rho, or by squeeze theorem. Is there any mathematical way to show that this goes to +infinity? We were told we were unable to evaluate limits by observation.
 
  • #7
Scrope said:
the way I had learned it was that the only way to prove a limit existed was through methods such as delta-epsilon definition, switching to R or Rho, or by squeeze theorem. Is there any mathematical way to show that this goes to +infinity? We were told we were unable to evaluate limits by observation.
For the record, here is the limit in the image link you posted:
$$\text{Find }\lim_{(x, y) \to (0, 1)} \tan^{-1}\left[\frac{x^2 + 1}{x^2 + (y - 1)^2}\right]$$
The problem asks you to find the limit, but doesn't ask you to prove that this value is the limit. Proving the limit would require the use of a delta-epsilon proof or a theorem such as the squeeze theorem.

A major difficulty with these kinds of limits is when the limit is an indeterminate form such as ##[\frac 0 0]##. This limit is not one of the indeterminate forms, since the numerator approaches 1 and the denominator approaches 0 through the positive numbers.
 
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  • #8
Scrope said:
the way I had learned it was that the only way to prove a limit existed was through methods such as delta-epsilon definition, switching to R or Rho, or by squeeze theorem. Is there any mathematical way to show that this goes to +infinity? We were told we were unable to evaluate limits by observation.

Yes, the way to do it is to use the definition of the notion ##(x,y) \to (0,1)##. Just using that definition (and a few simple manipulations), it is easy to show that for any (large) number ##N > 0##, we know that as ##(x,y)## nears ##(0,1)## (in a well-defined sense that you are supposed to know) we have that the denominator of the inside fraction becomes ##< 1/N##. The numerator is ## \geq 1##, so the inside fraction is ##> N##. That is essentially the definition of ##\text{fraction} \to +\infty.##
 
  • #9
Mark44 said:
For the record, here is the limit in the image link you posted:
$$\text{Find }\lim_{(x, y) \to (0, 1)} \tan^{-1}\left[\frac{x^2 + 1}{x^2 + (y - 1)^2}\right]$$
The problem asks you to find the limit, but doesn't ask you to prove that this value is the limit. Proving the limit would require the use of a delta-epsilon proof or a theorem such as the squeeze theorem.

A major difficulty with these kinds of limits is when the limit is an indeterminate form such as ##[\frac 0 0]##. This limit is not one of the indeterminate forms, since the numerator approaches 1 and the denominator approaches 0 through the positive numbers.
Oh, that's easier to read than the tiny image. I misread what x and y we're approaching.
 
  • #10
It doesn't matter whether the 'inside' is +∞ or -∞ , its tan-1 is π/2 in either case.

You appear to be able to approach the limit in several ways including from y>1.

The denominator is indeed the squared distance of a point (x, y) from (0,1).
 
  • #11
epenguin said:
It doesn't matter whether the 'inside' is +∞ or -∞ , its tan-1 is π/2 in either case.
Not so. ##\lim_{x \to -\infty}\tan^{-1}(x) = -\frac \pi 2##. For this inverse trig function, the restricted-domain function ##y = \tan(x), -\frac \pi 2 < x < \frac \pi 2## is used.
 

What is a multi-variable limit?

A multi-variable limit is the value that a function approaches as the inputs, or variables, approach a particular point. This is similar to a regular limit, but with multiple variables instead of just one.

How do you solve a multi-variable limit?

To solve a multi-variable limit, you need to approach the point from different paths or directions. This is called taking the limit along different paths. If the limit exists along all paths, it is equal to the same value regardless of the path taken. If the limit does not exist along all paths, it is called a multi-variable limit DNE (does not exist).

What tools are needed to solve a multi-variable limit?

To solve a multi-variable limit, you will need knowledge of basic algebra, trigonometry, and calculus. You will also need to understand the concept of limits and how they apply to functions with multiple variables.

Why is solving multi-variable limits important?

Solving multi-variable limits is important because it helps us understand the behavior of functions in multiple dimensions. It is also a fundamental concept in calculus and is used to solve more complex problems in fields such as physics, engineering, and economics.

What are some common strategies for solving multi-variable limits?

Some common strategies for solving multi-variable limits include using algebraic manipulation, factoring, and substitution. Another strategy is to graph the function and visually analyze the behavior near the point in question. Additionally, you can use the properties of limits, such as the limit laws, to simplify the problem and find the limit.

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