Math Problem Statement

Using: limx→5f(x)=6 and limx→5g(x)=4, evaluate limx→5f(x)+g(x)6f(x). Limit =

Solution

To evaluate the limit of the expression limx5f(x)+g(x)6f(x)\lim_{x \to 5} \frac{f(x) + g(x)}{6f(x)}, we can apply the properties of limits, which allow us to evaluate the limit of the numerator and denominator separately.

Given: limx5f(x)=6andlimx5g(x)=4\lim_{x \to 5} f(x) = 6 \quad \text{and} \quad \lim_{x \to 5} g(x) = 4

Now, applying the limits to the expression:

limx5f(x)+g(x)6f(x)=limx5(f(x)+g(x))limx56f(x)\lim_{x \to 5} \frac{f(x) + g(x)}{6f(x)} = \frac{\lim_{x \to 5} (f(x) + g(x))}{\lim_{x \to 5} 6f(x)}

Step 1: Evaluate the numerator:

limx5(f(x)+g(x))=limx5f(x)+limx5g(x)=6+4=10\lim_{x \to 5} (f(x) + g(x)) = \lim_{x \to 5} f(x) + \lim_{x \to 5} g(x) = 6 + 4 = 10

Step 2: Evaluate the denominator:

limx56f(x)=6limx5f(x)=66=36\lim_{x \to 5} 6f(x) = 6 \cdot \lim_{x \to 5} f(x) = 6 \cdot 6 = 36

Step 3: Combine the results:

1036=518\frac{10}{36} = \frac{5}{18}

Thus, the limit is: 518\boxed{\frac{5}{18}}

Would you like further details on any part of the process?


Here are 5 related questions to explore:

  1. How do you apply limit laws to more complex expressions?
  2. What happens if a limit involves indeterminate forms, like 00\frac{0}{0}?
  3. How do you handle limits involving infinity?
  4. Can L'Hopital's Rule be applied to this problem, and why or why not?
  5. What if the limits for f(x)f(x) and g(x)g(x) approached infinity?

Tip: Always check if the limit form is indeterminate before deciding on an evaluation method.

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Math Problem Analysis

Mathematical Concepts

Calculus
Limits
Limit Laws

Formulas

lim(x → a) [f(x) + g(x)] = lim(x → a) f(x) + lim(x → a) g(x)
lim(x → a) [cf(x)] = c * lim(x → a) f(x)

Theorems

Limit laws

Suitable Grade Level

Grades 11-12 (Calculus)