lim *x*→2 *f*(*x*) = 1    lim *x*→2 *g*(*x*) = −5    lim *x*→2 *h*(*x*) = 0,

find im *x*→2 [*f*(*x*) + 5*g*(*x*)]

We are tasked with finding the following limit:

\[
\lim_{x \to 2} \left[ f(x) + 5g(x) \right]
\]

We are given:
- \(\lim_{x \to 2} f(x) = 1\)
- \(\lim_{x \to 2} g(x) = -5\)

Now, apply the sum and scalar multiple properties of limits:

1. The limit of a sum of functions is the sum of their limits:
   \[
   \lim_{x \to 2} \left[ f(x) + 5g(x) \right] = \lim_{x \to 2} f(x) + \lim_{x \to 2} 5g(x)
   \]

2. The limit of a constant multiplied by a function is the constant multiplied by the limit of the function:
   \[
   \lim_{x \to 2} 5g(x) = 5 \cdot \lim_{x \to 2} g(x) = 5 \cdot (-5) = -25
   \]

Now substitute the known values:
\[
\lim_{x \to 2} f(x) + \lim_{x \to 2} 5g(x) = 1 + (-25) = -24
\]

Thus, the limit is:
\[
\lim_{x \to 2} \left[ f(x) + 5g(x) \right] = -24
\]

Would you like more details on the steps? Here are some related questions:

1. How do we handle limits when a function involves multiplication instead of addition?
2. What is the formal proof for the sum and product rules of limits?
3. How do we calculate limits when they involve indeterminate forms like \(0/0\)?
4. How does L'Hôpital's rule apply to more complex limit problems?
5. Can you explain how limits behave at infinity for rational functions?

**Tip:** When handling limits, remember that linear operations like addition and scalar multiplication can often be separated, simplifying calculations!

lim x→2 f(x) = 1, lim x→2 g(x) = −5, lim x→2 h(x) = 0, find lim x→2 [f(x) + 5g(x)]

Discover how to calculate the limit of f(x) + 5g(x) as x approaches 2, using the sum rule and scalar multiplication properties of limits. This example provides a step-by-step guide, ideal for high school calculus students.

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