The function 1/x^21x2 is always positive, and it increases without bound as x gets close to 00 and decreases toward zero as x gets bigger and bigger (whether positive or negative.)
You might know its graph:
graph{y=1/x^2 [-14.06, 14.42, -3.99, 10.25]}
Now in this question, the function is f(x) = 5-1/x^2f(x)=5−1x2.
Starting with 55, we will subtract 1/x^21x2 which is a number greater than 00, but with no upper bound.
That means at for some xx we will subtract 500500, and for some other xx we will subtract 5,0005,000 and for another subtract 60,00060,000 from 55. (We will never subtract a negative, because 1/x^21x2 is never negative.)
The result of this subtraction will be at most 55 but for very large positive 1/x^21x2, we will get very 'big' negatives, like -100−100 and then -10,000−10,000 and so on.
So, there is no lower bound for ff.
Upper bounds
f(x)f(x) is bounded above (by every number greater than or equal to 55) but is not bounded below. 55 is the least upper bound.
