The problem involves evaluating the limit of the expression [sin (x^2 + y^2 + z^2)] / [(x^2 + y^2 + z^2)^(1/2)] as the variables x, y, and z approach 0. The subject area pertains to multivariable calculus and limits.
There are multiple attempts to analyze the limit, with some participants expressing uncertainty about their approaches. Guidance has been offered regarding the substitution and manipulation of the expression, but no consensus has been reached on the correctness of the methods discussed.
Participants indicate that this is a homework assignment, and there is a sense of seeking clarification and support in understanding the problem. Some express gratitude for responses received, highlighting a collaborative atmosphere.
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].