As you say, with [itex]u= x^2+ y^2+ z^2[/itex],iamyes said:Homework Statement
Determine whether the following limit exist. If so, find its value
[sin (x^2 + y^2 + z^2 )] / [(x^2 + y^2 + z^2)^1/2)]
as x,y,z approach 0,0,0
The Attempt at a Solution
i tried to do using the limit of sin Ѳ / Ѳ as Ѳ approaching 0
im from malaysia and this is my assignment question and i can't figure out the way to solve this question
HallsofIvy said:As you say, with [itex]u= x^2+ y^2+ z^2[/itex],
[tex]\frac{sin(x^2+y^2+ z^2)}{(x^2+ y^2+ z^2)^{1/2}}= \frac{sin(u^2)}{u}[/tex]
Multiply both numerator and denominator by u to write that as
[tex]u\frac{sin(u^2)}{u^2}[/tex]
and now use [itex]sin(\theta)/\theta[/itex].
A 3 variable limit problem refers to a mathematical problem where three variables (usually x, y, and z) are involved in determining the limit of a function as one or more of the variables approach a certain value.
To solve a 3 variable limit problem, you need to first identify the variables and the function involved. Then, you can use techniques such as substitution, factoring, or L'Hopital's rule to simplify the expression and evaluate the limit.
Studying 3 variable limit problems is important because it helps us understand the behavior of a function as multiple variables approach a certain value. This is useful in fields such as physics, engineering, and economics, where multiple variables are often involved in real-world problems.
Yes, there are several techniques and strategies that can be used to solve 3 variable limit problems. Some of these include using algebraic manipulation, applying the squeeze theorem, or using graphical methods such as plotting the function on a 3D graph.
Yes, some common mistakes to avoid when solving 3 variable limit problems include forgetting to check for discontinuities or using incorrect substitution values. It is also important to double-check your algebraic manipulations and to use the correct limit laws and theorems.