Question

Let f be some function for which you know only that

if 0<|x−3|<2, then |f(x)−7|<1.

Select the TRUE statement below.

(a) lim f(x) = 7.

x→3

(b) If|x−3|<2,then|f(x)−7|<1.

(c) If0<|x−3|<1,then|f(x)−7|<1.

(d) If0<|x−3|<1,then|f(x)−7|<12.

Answer 1

Step-by-step explanation: The correct statement is:

(c) If 0 < |x - 3| < 1, then |f(x) - 7| < 1.

This is true based on the given information about the function. The statement matches the condition that if the absolute value of the difference between x and 3 is between 0 and 1, then the absolute value of the difference between f(x) and 7 is less than 1.

Option (a) is not necessarily true because we do not have enough information to determine the limit of f(x) as x approaches 3.

Option (b) is not necessarily true because the given information only specifies the condition for the range 0 < |x - 3| < 2, not for all values of |x - 3| < 2.

Option (d) is not true because it states that |f(x) - 7| < 12, which is not supported by the given information that states |f(x) - 7| < 1.