Let's solve the limit:
[tex]\[
\lim_{x \to a} \frac{3f(x)}{h(x) + 9f(x)}
\][/tex]
Step-by-step solution:
1. Evaluate the numerator:
The numerator of the fraction is [tex]\( 3f(x) \)[/tex]. As [tex]\( x \)[/tex] approaches [tex]\( a \)[/tex]:
[tex]\[
\text{Numerator} = 3f(x) \rightarrow 3f(a)
\][/tex]
2. Evaluate the denominator:
The denominator of the fraction is [tex]\( h(x) + 9f(x) \)[/tex]. As [tex]\( x \)[/tex] approaches [tex]\( a \)[/tex]:
- The term [tex]\( h(x) \)[/tex] approaches [tex]\( h(a) \)[/tex]
- The term [tex]\( 9f(x) \)[/tex] approaches [tex]\( 9f(a) \)[/tex]
Therefore, the denominator is:
[tex]\[
\text{Denominator} = h(x) + 9f(x) \rightarrow h(a) + 9f(a)
\][/tex]
3. Combine the results:
Putting the evaluated results together, we get:
[tex]\[
\lim_{x \to a} \frac{3f(x)}{h(x) + 9f(x)} = \frac{3f(a)}{h(a) + 9f(a)}
\][/tex]
So, the limit is:
[tex]\[
\boxed{\frac{3f(a)}{h(a) + 9f(a)}}
\][/tex]
