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1. ### Is it possible for 2 tangent lines of this function to be perpendicular?

Your constants are wrong. Also, that .5x should be a -.5x. You want to find a constant such that 2*x + b1 = x^2 + 2 for x = 2. You also want to find a constant such that -0.5*x + b2 = x^2 + 2 for x = -0.25
2. ### Is it possible for 2 tangent lines of this function to be perpendicular?

Well, f'(1) is the slope of the tangent line to f(x) at x = 1, and f'(-0.25) is the slope of the tangent line to f(x) at x = -0.25. So, you are looking for two lines: y = a1x + b1 and y = a2x + b2 That are tangent to f(x) at x=x1 and x = x2, respectively, but which are...
3. ### Is it possible for 2 tangent lines of this function to be perpendicular?

Yes, exactly!
4. ### Is it possible for 2 tangent lines of this function to be perpendicular?

TO THE OP: I think you may be misunderstanding the question you were asked. The question is: find two tangent lines to f(x) such that the two are perpendicular to each other. In other words, they will both be parallel to the function f(x) at their respective points, but they will be...
5. ### Simple probability proof about limits

I'll give you a hint for part (a): Let {En} be a sequence of sets. For each n, let Cn be the compliment of En. Suppose that P(C1) = a, P(C2) = a/2, P(C3) = a/4, etc. In other words, P(Cn) = a/2(n-1). What can we say then about P(\bigcup Cn)? Remember that 1/2n is a geometric series. Tell me...