If f is a function such that the limit as x approaches a of the quotient of the quantity f of x minus f of a and the quantity x minus a equals 7, then which of the following statements must be true?
[f(x)-f(a)] /(x-a) = 7 as x-->a
is the definition of the derivative
f'(a) = 7
f'(a) = 7
Well, if the limit of that quotient equals 7, it means that at the point a, f(x) - f(a) is "sensitive" to changes in x. In other words, it means that f(x) is quite the emotional function - it gets all worked up and starts "crying" every time someone tries to subtract a from x. So, to answer your question, the statement that must be true is that f(x) is an expert at shedding function tears.
The given information can be written as:
lim(x->a) (f(x) - f(a)) / (x - a) = 7
To determine which statements must be true, we'll consider some possibilities step-by-step:
1. If f(x) is a constant function, meaning f(x) = c for all x, then the difference f(x) - f(a) would always be zero. This would make the denominator (x - a) non-zero, leading to a contradiction with the given limit being 7. Therefore, f(x) cannot be a constant function.
2. If f(x) is a linear function, meaning f(x) = mx + b for some constants m and b, then the difference f(x) - f(a) would be mx + b - ma - b = m(x - a). In this case, the limit can be written as:
lim(x->a) m(x - a) / (x - a) = m
Since the given limit is 7, we can conclude that m = 7. Therefore, if f(x) is a linear function, it must have a slope of 7.
3. If f(x) is not a linear function, it could be any other type of function. In this case, we cannot determine the exact value of the limit without additional information.
Based on the analysis, the only statement that must be true is that f(x) must have a slope of 7 if it is a linear function.
To determine which of the following statements must be true, let's first analyze the given information.
The given limit states that as x approaches a, the quotient of (f(x) - f(a)) divided by (x - a) equals 7. In other words, the rate of change of the function f(x) as x approaches a is 7.
Now, let's consider the statements and determine which must be true:
1. f(a) = 7
This statement does not necessarily have to be true. The limit tells us about the rate of change of the function as x approaches a, but it doesn't give us any information about the value of f(a) itself. So, f(a) could be any value; it doesn't have to be 7.
2. f is continuous at x = a
This statement must be true. From the given limit, we know that the rate of change of f(x) as x approaches a is 7. For this to be the case, f(x) must be continuous at x = a. If f(x) had a discontinuity at x = a, then the rate of change would not be well-defined.
3. f'(a) = 7
This statement is not necessarily true. The given limit only tells us the rate of change of f(x) as x approaches a; it does not imply that f'(a) itself is equal to 7. The derivative of f at x = a could be any value; it doesn't have to be 7.
Therefore, the only statement that must be true is:
- f is continuous at x = a.