To find the limit of the expression [tex]\(\frac{5x^2 - 3x}{9x}\)[/tex] as [tex]\(x\)[/tex] approaches [tex]\(0\)[/tex], we'll follow a step-by-step approach:
1. Given Expression:
[tex]\[
\lim_{x \to 0} \frac{5x^2 - 3x}{9x}
\][/tex]
2. Simplifying the Expression:
We can start by simplifying the fraction inside the limit. Notice that both the numerator and the denominator have a common factor of [tex]\(x\)[/tex].
[tex]\[
\frac{5x^2 - 3x}{9x} = \frac{x(5x - 3)}{9x}
\][/tex]
3. Cancel Out the Common Factor:
Since [tex]\(x \neq 0\)[/tex] (we're considering the limit as [tex]\(x\)[/tex] approaches 0, but not equal to 0), we can safely cancel [tex]\(x\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{x(5x - 3)}{9x} = \frac{5x - 3}{9}
\][/tex]
4. Taking the Limit:
Now, we need to find the limit of the simplified expression as [tex]\(x\)[/tex] approaches 0:
[tex]\[
\lim_{x \to 0} \frac{5x - 3}{9}
\][/tex]
As [tex]\(x\)[/tex] approaches 0, the term [tex]\(5x\)[/tex] will also approach 0. Therefore, the expression inside the limit simplifies to:
[tex]\[
\frac{5(0) - 3}{9} = \frac{-3}{9} = -\frac{1}{3}
\][/tex]
5. Conclusion:
Thus, the limit of [tex]\(\frac{5x^2 - 3x}{9x}\)[/tex] as [tex]\(x\)[/tex] approaches 0 is:
[tex]\[
\boxed{-\frac{1}{3}}
\][/tex]
