Step 1a

To find the limit of the function \$f(x) = 4x^2 - 7x + 5\$ as x approaches 2, we can directly substitute the value 2 into the function: f(2) = \$4(2)^2 - 7(2) + 5\$ = 4(4) - 14 + 5 = 16 - 14 + 5 = 7.

Therefore, the limit of the function \$f(x) = 4x^2 - 7x + 5\$ as x approaches 2 is 7.

Explanation: After inputting the value into the function, the result is multiplied by seven.

Step 2a

To determine continuity, we need to check three conditions:

Explanation: Here, let’s introduce and discuss three conditions.

Step 2b

Let’s evaluate each condition:

1. The function f(x) = 3x - 2 is defined for all real numbers, including x = 4. Therefore, the function is defined at x = 4.

Explanation: Here, use the first condition to satisfy the given function.

Step 2c

2. To find the limit as x approaches 4, we substitute x = 4 into the function: \$ \lim_{x \to 4} 3x - 2 = 3(4) - 2 = 12 - 2 = 10 \$.

Thus, the limit of f(x) as x approaches 4 is 10.

Explanation: Here, use the second condition to satisfy the function.

Step 2d

3. Now, we compare the value of the function at x = 4 with the limit: f(4) = 3(4) - 2 = 12 - 2 = 10.

The value of the function f(x) at x = 4 is equal to the limit.

Since all three conditions are satisfied, we can conclude that the function f(x) = 3x - 2 is continuous at x = 4.

Explanation: Here, satisfying the third condition, we conclude that f(x) = 3x - 2 is continuous at x = 4.