Final answer:
The correct answer is option (b) because it describes a function that increases on both sides of x=1 and approaches the limit of 5 as x gets close to 1, fulfilling the conditions given.
Step-by-step explanation:
If the function g is increasing for x less than 1 and greater than 1, and the limit of g(x) as x approaches 1 is 5, we can eliminate option (a) g(x)=5 for all x because this suggests that g is constant, not increasing. Option (d) g(x)=x+4 for x ≠1 would represent a function that is always increasing, including at x=1, which contradicts the limit as x approaches 1. Therefore, we are left with scenarios where the behavior of g changes around x=1, which are options (b) and (c).
Since the limit of g(x) as x approaches 1 is equal to 5, the function value must approach 5 from both sides as x gets close to 1. This means that g must be below 5 just before reaching x=1 and above 5 just after. Therefore, the correct answer is option (b) g(x)<5 for x<1, g(x)>5 for x>1 as it accurately describes a function that is increasing on both sides of x=1 with a limit of 5 as x approaches 1.