Limits of fractions of polynomials and trig functions

In summary, the conversation discusses two different limit problems, one with the equation \lim_{x \rightarrow 1} \frac{1-cosx}{x^2} and the other with the equation \lim_{x \rightarrow 3} \frac{\sqrt{x^2-6x+9}}{x-3}. The conversation includes attempts at solving these problems, including plugging in values and simplifying equations. It also mentions a potential typo in the first problem.
  • #1
carbz
35
0
I have two...

Homework Statement


The the limit


Homework Equations


[itex]\lim_{x \rightarrow 1} \frac{1-cosx}{x^2}[/itex]


The Attempt at a Solution


I figured to just plug in 1, but I wanted to make sure...

Homework Statement


Find the limit


Homework Equations


[itex]\lim_{x \rightarrow 3} \frac{\sqrt{x^2-6x+9}}{x-3}[/itex]


The Attempt at a Solution


I plugged in the 3, and got 3/0, then I got lost...
 
Physics news on Phys.org
  • #2
In 2, did you try simplifying the numerator? (What are the roots of the polynomial?)
 
  • #3
Yes, I tried doing that.

[itex](x-3)(x-3)[/itex]

However, I forgot how to get rid of that radical. Squaring wouldn't work, so I have no idea.


Also, no thoughts on the first one?
 
  • #4
What is a short hand expression for (x-3)(x-3)?
 
  • #5
(x-3}^2. Oh yeah, so that takes away the square root, and after everything, it leaves 0. thank you.
 
  • #6
In the first one, are you sure the problem isn't x->0 instead of x->1 ?
 
  • #7
it is 1, not 0.
 
  • #8
Well, then your book has a typo..
 
  • #9
it's not from my book, it was my teacher.
 
  • #10
Well, it's 99% certain that your teacher meant to write 0 instead of 1. With 1, it's trivial, since both the numerator and denominator are finite, nonzero constants in that limit.
 
  • #11
Then he either blundered, or tried to fool you.

Your function is defined&continuous on all values of x except x=0.

Your original approach is perfetly valid in the case of x=1.
 
  • #12
allright, thankyou.
 

Related to Limits of fractions of polynomials and trig functions

1. What is a limit of a fraction of polynomials and trig functions?

A limit of a fraction of polynomials and trig functions is the value that a function approaches as its input (x-value) approaches a certain value. It can also be thought of as the value that a function gets closer and closer to, but does not necessarily reach, as the input gets closer to a specific value.

2. How do you find the limit of a fraction of polynomials and trig functions?

To find the limit of a fraction of polynomials and trig functions, you can either evaluate the function at the specific value that the input is approaching, or use algebraic techniques such as factoring or simplifying to determine the limit. You can also use the rules of limits, such as the sum, difference, product, and quotient rules, to find the limit.

3. Can a limit of a fraction of polynomials and trig functions be undefined?

Yes, a limit of a fraction of polynomials and trig functions can be undefined. This can happen if the function has a vertical asymptote, meaning the function approaches positive or negative infinity as the input approaches a certain value. It can also be undefined if the function has a point discontinuity, where the limit from the left and the limit from the right are not equal.

4. What are the common types of limits of fractions of polynomials and trig functions?

The common types of limits of fractions of polynomials and trig functions include horizontal asymptotes, where the limit approaches a constant value, and slant asymptotes, where the limit approaches a linear function. Other types of limits include removable discontinuities, infinite limits, and oscillating limits.

5. Why are limits of fractions of polynomials and trig functions important in mathematics?

Limits of fractions of polynomials and trig functions are important in mathematics because they allow us to understand the behavior of functions at specific points or as the input approaches certain values. They are also essential in calculus, as they are used to define derivatives and integrals, which are fundamental concepts in the study of functions and their properties.

Similar threads

Find limit of multi variable function
    • Calculus and Beyond Homework Help
Replies
10
Views
856
Evaluate the limit of this harmonic series as n tends to infinity
    • Calculus and Beyond Homework Help
Replies
17
Views
700
Radii of stacked circles inside the graph of y = |x|^1.5
    • Calculus and Beyond Homework Help
Replies
10
Views
890
Problems solving a limit which results in an indeterminate form
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Limit and Integration of ##f_n (x)##
    • Calculus and Beyond Homework Help
Replies
8
Views
734
Limit problem involving two circles and a line
    • Calculus and Beyond Homework Help
Replies
8
Views
847
Existence of directional derivative
    • Calculus and Beyond Homework Help
Replies
4
Views
593
Calculate the limit cos(x)/sin(x) when x approaches 0
    • Calculus and Beyond Homework Help
Replies
12
Views
921
Find limit involving square of sine
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Why does the numerator become 1 in the root test for solving series?
    • Calculus and Beyond Homework Help
Replies
7
Views
1K

Hot Threads

Recent Insights

Back
Top