Prove the following equation

Question

𝑈𝑠𝑒 𝑡ℎ𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑡𝑜 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 lim

𝑛→−5

1

(𝑥+5)

6 = ∞

Answer 1

I'm guessing that you meant something like this, it's a little hard to tell from the formatting of your question.

As you can see, directly plugging in -5 for x gives you 1/0, which is equal to infinity. It may be hard to understand why that is. Think about it like this, as you divide a number by smaller and smaller values the fraction gets bigger, 1/0.5 = 2, 1/0.1 = 10. 1/0.0002 = 5000, and so on. In this case you can think of 0 as in infinitely small number, and since dividing 1 by smaller numebers equals bigger values dividing by an infinitely small number equals infinity.

You may be tempted to use l'Hopital's rule, but that is not allowed in this case. That only works when the limit is indeterminate, when it is 0/0 or infinity/infinty, since this causes the numerator and denominator to cancel.