To evaluate the limit

$\lim_{(x, y, z) \to (0, 0, 0)} \frac{\sin(x^2 + y^2 + z^2)}{x^2 + y^2 + z^2},$

let’s proceed step by step:

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Step 1: Substitution and simplification

Let $u = x^2 + y^2 + z^2$. When $(x, y, z) \to (0, 0, 0)$, we also have $u \to 0$.

The expression becomes: $\lim_{u \to 0} \frac{\sin(u)}{u}.$

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Step 2: Recognizing a standard limit

The standard limit in calculus is: $\lim_{u \to 0} \frac{\sin(u)}{u} = 1.$

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Step 3: Conclusion

Thus, the original limit is: $\lim_{(x, y, z) \to (0, 0, 0)} \frac{\sin(x^2 + y^2 + z^2)}{x^2 + y^2 + z^2} = 1.$

Would you like further explanations or detailed steps on how to handle such limits?

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Related Questions:

1. How is the substitution $u = x^2 + y^2 + z^2$ chosen in multivariable calculus?
2. What is the proof for the standard limit $\lim_{u \to 0} \frac{\sin(u)}{u} = 1$?
3. How can polar or spherical coordinates be used to evaluate multivariable limits?
4. Are there situations where substitution doesn't work for evaluating multivariable limits?
5. How does this limit behave if the numerator is replaced with $\cos(x^2 + y^2 + z^2)$?

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Tip:

When working with limits involving trigonometric functions, always check if standard limits can simplify the problem.