គណនាលីមីត : 
 Lim_x->3[ ( ⁵√(3x+10-1))/(x³ + x² -x + 15) 
 Please tell me 

To find the limit \(\lim_{{x \to 3}} \left( \frac{\sqrt[5]{3x + 10 - 1}}{x^3 + x^2 - x + 15} \right)\), let's go through the steps carefully:

### 1. Substitute \(x = 3\) into the expression.
First, substitute \(x = 3\) directly into the expression to check if the function is well-defined at \(x = 3\).

\[
\text{Numerator} = \sqrt[5]{3(3) + 10 - 1} = \sqrt[5]{9 + 10 - 1} = \sqrt[5]{18}
\]

\[
\text{Denominator} = 3^3 + 3^2 - 3 + 15 = 27 + 9 - 3 + 15 = 48
\]

So, the expression simplifies to:

\[
\frac{\sqrt[5]{18}}{48}
\]

### 2. Evaluate the limit:
Since substituting \(x = 3\) gives a finite value, the limit is simply the value of the expression when \(x = 3\).

\[
\lim_{{x \to 3}} \left( \frac{\sqrt[5]{3x + 10 - 1}}{x^3 + x^2 - x + 15} \right) = \frac{\sqrt[5]{18}}{48}
\]

Therefore, the limit is \(\frac{\sqrt[5]{18}}{48}\).

### Would you like more details or have any questions?

Here are 8 related questions you might consider:
1. How do you calculate the fifth root of a number?
2. What is the significance of limits in calculus?
3. Can limits always be computed by direct substitution?
4. What happens if substituting a limit gives an indeterminate form like \(\frac{0}{0}\)?
5. How do you approach limits involving complex functions?
6. What is L'Hôpital's Rule, and when can it be applied?
7. How do you handle limits that go to infinity?
8. What are the common techniques to simplify limit expressions?

**Tip:** When evaluating limits, always check if direct substitution works first; if not, consider algebraic manipulation, factorization, or L'Hôpital's Rule.

Learn how to calculate the limit \(\lim_{{x \to 3}} \left( \frac{{\sqrt[5]{{3x + 10 - 1}}}}{{x^3 + x^2 - x + 15}} \right)\) step-by-step with a detailed explanation. This problem involves algebraic expressions, roots, and limits. Suitable for college-level mathematics.

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