Final answer:
To make the function f continuous at x = 3, we need to ensure that the limit of f(x) as x approaches 3 from both sides is equal. In this case, there is no value of a that will make f continuous at x = 3.
Step-by-step explanation:
To make the function f continuous at x = 3, we need to ensure that the limit of f(x) as x approaches 3 from both sides is equal. In this case, we need to find the value of a such that f(x) is defined and the limit of f(x) exists as x approaches 3.
Let's calculate the limits:
Limit as x approaches 3 from the left side (denoted as x → 3-):
lim x → 3- f(x) = lim x → 3- x/(x² - 9)
Limit as x approaches 3 from the right side (denoted as x → 3+):
lim x → 3+ f(x) = lim x → 3+ x/(x² - 9)
To calculate these limits, substitute x = 3 into the expression and simplify:
lim x → 3- f(x) = lim x → 3- 3/(3² - 9) = lim x → 3- 3/0 which is undefined.
lim x → 3+ f(x) = lim x → 3+ 3/(3² - 9) = lim x → 3+ 3/6 = 1/2.
Since the limits from the left and right sides are not equal, the function f(x) is not continuous at x = 3 for any value of a.