Esteem the function and calculate limit left and right

Question

Esteem the function f(x)=1/(x³-1) for near values of 1 and then calculate them lim of 1/(x³-1) when x->1- and lim of 1/(x³-1) when x->1+.

Answer 1

I believe that the verb esteem is used as a synonym for estimate.

 

So if you're trying to find the limit from the left, you would plug in .99 or .95 or .9 . Once you get the hang of asymptotic behavior, you really just need to find out if this number is positive or negative.

 

So f(.9) = -3,69

    f(.95) = -.70114

    f(.99) = -33.669

So as x is approaching 1 from the left, the numbers less than 1 but almost 1, f(x) approaches -infinity.

 

So if you want to find the limit from the right, x-> 1⁺, you need to try 1.1, 1.01, 1.001.

 

f(1.1) = 3.021

f(1.01) = 33.002

f(1.001) = 333.000222

 

So the limit as x -> 1⁺ = infinity.

 

Hope this helped

Answer 2

A very important calculus question! Although, I do think you mean "estimate"

 

To estimate this function for values near to 1 we need to actually calculate it for a few values close to 1. Like 0.99. 0.9, 1.1, and 1.01. Then do something similar to that to figure out the left and right limits.

 

So what is f(0.9)?

 

f(0.9) = 1 / (0.9^3 - 1) = 1 / (0.729 - 1) = 1 / -0.271 = -3.69

 

What about f(0.99)?

 

f(0.99) = 1 / (0.99^3 -1) = 1 / (0.97 - 1) = 1 / -0.03 = -33.333

 

Do some similar calculations and you should get that

 

f(1.01) = 33.33

 

f(1.1) = 3.02

 

 

So, what is the limit when x -> 1-? I.e. what is the limit as x approaches 1 from the left? 

 

Well, we saw that f(0.9) was about -3.5. And f(0.99) was about -33. If you do one more calculation you should find that f(0.999) = -333.33. As you move x closer and closer to 1, you will find f(x) just gets bigger and bigger in the negative. Thus,

 

Limit of f(x) as x->1- = -infinity. 

 

You can also figure this out by talking through it, without really calculating values. Just imagine x for a value really close to 1 like 0.9999999999. You cube that and you get a number that is similarly close to 1, you basically just get 0.9999999 again. Now, what is 0.999999999 - 1? It's an incredibly small negative number. Okay, so what happens when you divide 1 by an incredibly small, negative number? You get a really big negative number. Cool, so that's why the limit is negative infinity. 

 

You should be able to follow similar procedure to calculate the limit as x->1+. Just look at f(1.1), then f(1.01), and then ask yourself what happens at values *really* close to 1 but greater than 1, like 1.000000000001. Be careful with your negative signs and you will get it!