# A couple of limit computations

• flipsvibe
In summary, the students are struggling to solve the homework equations. They are not familiar with the Taylor series, and do not know how to solve the equations without using l'hospitals rule. If they were to continue working on the equations, they would first be able to solve for x in the numerator, and then use the limit to find the limit as x approaches 0.
flipsvibe

## Homework Statement

Compute the limits. If they don't exist, then explain.

1) lim as x approaches 0 [cos(x) - 1] / [sin^2(x) + x^3]

2) lim as x approaches 0 √(x^2 + x^3) / (x + 2x^2)

## The Attempt at a Solution

1) I replaced the sin^2x in the denominator with 1 - cos^2x and tried to work from there, but I couldn't get anything to cancel.

2) I couldn't figure out anything on this one either. I factored out an x in the denominator, but again, that didn't lead anywhere.

We are doing a semester review for the final, and I am starting to get a little nervous, because I don't remember struggling with limit computations when we learned them. Any help that can get me going the right direction would be greatly appreciated. Thank you.

Edit: This course was taught integrals first, then derivatives. We have not learned L'hospitals rule yet.

Last edited:
for the second one, did you factor out an x in the numerator?
If not, that would greatly help
knowing l'hopital's rule would also really help for the first one

i know you can't use l'hopital's rule, but i get -1/2 for the first one using l'hopital's rule

Have you learned the Taylor series? I'm guessing "no", but your course seems to be ordered in a weird way.

ideasrule said:
Have you learned the Taylor series? I'm guessing "no", but your course seems to be ordered in a weird way.

That's the same thing i thought. I first learned about derivatives(Calc I) then about integrals(Calc II)

Thanks, I figured out the first one. I factored out an x^2 in the numerator. Giving me
√[x^2 (1+x)] / x(1+2x)
==> x √(1+x) / x(1+2x)
==> lim x --> 0 is equal to 1

that's what I got for the second one.
so for the first one, i have no clue how to do it without using l'hopital's rule
did you make any progress on that one?

That's the way I learned them in high school: derivatives -> integrals. But we are using apostol's calculus, and he goes in the order they were discovered historically.

No progress on the first one at all. I might just try to use L'hopital's rule anyways. We stopped for the semester one chapter before the book goes over it.

flipsvibe said:
That's the way I learned them in high school: derivatives -> integrals. But we are using apostol's calculus, and he goes in the order they were discovered historically.

cool, never heard of it, but the only problem I see in the first one is that x^3.
If that wasn't there, I probably could have applied the limit in about two to three steps

## 1. What is a limit computation?

A limit computation is a mathematical process used to determine the behavior of a function as the input approaches a certain value. It involves evaluating the function at values that are increasingly closer to the specified input and observing the resulting output values.

## 2. Why are limit computations important?

Limit computations are important because they allow us to understand the behavior of a function at certain points, even if the function is not defined at those points. They are also essential in calculus, as they are used to define derivatives and integrals.

## 3. How do you perform a limit computation?

To perform a limit computation, you first need to identify the input value that the function is approaching. Then, you can use algebraic techniques such as factoring, simplifying, and cancelling to manipulate the function into a form that is easier to evaluate. Finally, you can use substitution or direct evaluation to determine the limit.

## 4. What are the different types of limits?

There are three types of limits: one-sided limits, two-sided limits, and infinite limits. One-sided limits are used when the function approaches a specific input value from only one direction. Two-sided limits are used when the function approaches a specific input value from both directions. Infinite limits occur when the function grows or shrinks without bound as the input approaches a certain value.

## 5. When are limit computations not applicable?

Limit computations are not applicable when the function is not continuous at the specified input value. This means that the function has a jump, hole, or vertical asymptote at that point. In these cases, the limit does not exist, and other methods must be used to understand the behavior of the function.

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