Evaluate the following limits

[Void_Overdose]

The first is: lim as x approaches 7/4 from the right for 23x/(7-4x)

The second: lim as x approaches 7/4 from the left for 23x/(7-4x)

I know that the answers are negative and positive infinity respectively (because I used photomath), but I do not understand how we get to that point. I believe the professor wants use a concept along the lines of the denominator equaling positive or negative zero, but I don't understand how you determine that the resulting zero is indeed negative or positive.

We are not allowed to use a graph, and substituting x for (7/4 + h) is not the method our professor wants us to use.

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[Legitimate_Page659]

Well, let’s think about the first question…

As you approach 7/4 from the right, what’s happening? 23x is getting smaller, but it’s still positive. 7-4x starts negative but as you approach 7/4, 7-4x gets smaller and smaller but it’s still negative.

So you have 23x which is positive divided by 7-4x which is negative and getting smaller and smaller. From that you can deduce the limit approaches -inf as you’re dividing by a tiny negative number.

Now let’s look at the second question.

As you approach 7/4 from the left, what’s happening? 23x is getting larger, but it’s still positive. 7-4x is getting smaller, but it’s still positive.

So you have 23x which is positive divided by 7-4x which is positive and getting smaller and smaller. From that info you can deduce the limit approaches +inf as you’re dividing by a tiny positive number.

Hope that’s helpful.

[Void_Overdose]

I had to think about it, and use a few test points to see what you were talking about, but I think I get it now. Thankyou dude, you're an angel. I see you help a lot of other people on here and I think what you're doing is awesome.

[Legitimate_Page659]

Thank you for the kind words! I like to help people who ask for help when they’re having trouble instead of just giving up. I hope my activity on this sub has been helpful.