Question

Use limit laws to evaluate the following limit:

Assume limx→5f(x)=6 and limx→5g(x)=3, compute limx→5(f(x)+5g(x)+2)

Answer 1

The limit of the given expression as x approaches 5 is found by applying the sum rule for limits and multiplying the constant by the corresponding limits of f(x) and g(x), resulting in a limit of 23.

Step-by-step explanation:

To evaluate the limit of a sum of functions, we can use the sum rule for limits, which states that the limit of a sum is equal to the sum of the limits. Given that limx→5f(x) = 6 and limx→5g(x) = 3, we apply this rule to the given expression limx→5(f(x) + 5g(x) + 2).

We can break it down as follows:

• The limit of f(x) as x approaches 5 is 6.
• The limit of 5g(x) can be found by taking 5 times the limit of g(x) as x approaches 5, which is 5 × 3 = 15.
• The limit of a constant like 2 is just the constant itself, so limx→52 = 2.

When we combine these results using the sum rule, we get:

limx→5(f(x) + 5g(x) + 2) = limx→5f(x) + limx→55g(x) + limx→52

= 6 + 15 + 2

= 23

Therefore, the limit of the function as x approaches 5 is 23.

Answer 2

To evaluate the limit, use the limit laws to find the sum of the individual limits. In this case, the limit of (f(x) + 5g(x) + 2) as x approaches 5 is equal to the sum of the limits of f(x), 5g(x), and 2 as x approaches 5, which is 23.

Step-by-step explanation:

To evaluate the limit, we can use the limit laws. According to the limit laws, the limit of a sum is equal to the sum of the limits. Therefore, the limit of (f(x) + 5g(x) + 2) as x approaches 5 is equal to the sum of the limits of f(x), 5g(x), and 2 as x approaches 5.

1. Since limx→5f(x) = 6, the limit of f(x) as x approaches 5 is 6.
2. Similarly, limx→5g(x) = 3, so the limit of g(x) as x approaches 5 is 3.
3. Finally, the limit of 2 as x approaches 5 is simply 2.

We can now add the limits together to find the overall limit:

limx→5(f(x) + 5g(x) + 2) = 6 + 5(3) + 2 = 6 + 15 + 2 = 23.