Which of the following functions is continuous at x = 5? I put a for my answer, could someone please check this?
a. f(x)=(x^2-25)/(x+5)
b. (x^2-25)/(x-5), x cannot equal 5
20 when x equals 5
c.(x^2-25)/(x-5), x cannot equal 5
0 when x equals 5
d. all of the functions are continuous.
a) is the correct choice
To determine if a function is continuous at a specific point, we need to check if the function is defined at that point and if the limit of the function as x approaches that point exists and equals the value of the function at that point.
Let's analyze each option:
a. f(x) = (x^2 - 25)/(x + 5)
This function is discontinuous at x = -5 because the denominator becomes zero, making the function undefined at that point. Thus, it is not continuous at x = 5.
b. (x^2 - 25)/(x - 5)
This function is defined for all x except x = 5. However, we're interested in the continuity at x = 5. Since the function is not defined at x = 5, it is not continuous at that point.
c. (x^2 - 25)/(x - 5)
Similarly to option b, this function is not defined at x = 5. Consequently, it is not continuous at that point.
d. All of the functions are continuous.
Based on our analysis of options a, b, and c, we see that none of the functions are continuous at x = 5. Therefore, option d is incorrect.
Based on the information provided, it seems that none of the given functions are continuous at x = 5. Therefore, option a is not correct.
To determine if a function is continuous at a specific point, we need to check three things:
1. The function should be defined at that point.
2. The limit of the function as x approaches that point should exist.
3. The limit should equal the value of the function at that point.
Let's apply these criteria to each of the given functions at x = 5:
a. f(x) = (x^2 - 25)/(x + 5)
To check continuity at x = 5, we need to see if f(5) is defined. In this case, f(5) = 20. So, the function is defined at x = 5.
Next, we need to evaluate the limit of the function as x approaches 5. We can simplify f(x) as:
f(x) = (x - 5)(x + 5)/(x + 5)
= x - 5
Now, let's calculate the limit of f(x) as x approaches 5:
lim(x->5) f(x) = lim(x->5) (x - 5) = 5 - 5 = 0
Since the limit exists and is equal to the value of f(5), the function f(x) = (x^2 - 25)/(x + 5) is continuous at x = 5.
Now let's analyze the other options:
b. (x^2 - 25)/(x - 5), x cannot equal 5, f(5) = 20
Since the function is not defined at x = 5 in this case, it does not satisfy the first criterion for continuity. Hence, option b is not correct.
c. (x^2 - 25)/(x - 5), x cannot equal 5, f(5) = 0
Similar to option b, this function is not defined at x = 5, so it does not satisfy the first criterion for continuity. Thus, option c is not correct.
d. All the functions are continuous
From the analysis above, we can see that only option a satisfies all the criteria for continuity at x = 5. Therefore, the correct choice is a, and it seems you have made the right selection.