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Determine the infinite limit.

$ \displaystyle \lim_{x \to 5^+}\frac{x+1}{x-5} $

$\lim _{x \rightarrow 5^{+}} \frac{x+1}{x-5}=\infty$ since the numerator is positive and the denominator approaches 0 from the positive side as $x \rightarrow 5^{+}$

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So in this problem we are asked to evaluate the limits As X approaches five from the right hand side of X Plus one Over X -5. Okay, well first of all we know that as we're approaching five from the right hand side, these are numbers greater than five. And so our denominator is going to be some positive number because I have something greater than five, modest five. That's a positive number in the numerator. It's obviously a positive number because we're just adding one to it. Okay, so What else do we know we know that this denominator right here, X -5 is I'm approaching five from the right hand side. This number gets smaller and smaller and smaller and smaller. And so dividing by a smaller and smaller and smaller number, like that means to multiply by a larger and larger number Has this denominator approaches zero. So that means that this is really going to be positive infinity. And let's look at a graph, here's our graph, X plus one Over X -5. And you notice the graph here right as X approaches five here. Okay. What happens? Well, this graph just keep continues to climb all the way up. Go toward positive infinity. Thus we have positive affinity for this limit

Oklahoma State University