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# Limit and intersection point of a function

Let be a continuous and strictly increasing function. Consider the line . Prove that:

1. .
2. has a unique intersection point with the line above.

Solution

1. As we can without loss of generality assume that , hence . Thus,

Similarly , we can prove that .

2. The function is continuous and strictly decreasing. It follows from question (i.) that . Since it follows that there exists an such that which is unique due to the monotony of .