Limit of a Sin Function

In summary, the conversation discusses a problem involving using calculus to simplify an expression. A first year math student suggests expanding the denominator and canceling out terms, but gets stuck. Other users suggest using a known limit to solve the problem. Ultimately, the correct solution is found and the conversation ends with gratitude.
  • #1
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Homework Statement



[PLAIN]http://www4d.wolframalpha.com/Calculate/MSP/MSP120619i0ea2h15bci31300005d9a3422292h43h4?MSPStoreType=image/gif&s=43&w=174&h=39 [Broken]

Homework Equations


The Attempt at a Solution



So Wolfram says to use L'Hopital as the first step, we haven't learned anything about this yet so there has to be another way using calculus that a first year math student would know.

I would first expand the denominator of x^2-4 to (x+2)(x-2) and cancel out the x+2 in both numerator and denominator. And I'm left with:
((x-1)sin1)/(x-2)
This is where I get lost...

Any help would be much appreciated!
Jim
 
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  • #2
One cannot separate (x+2) from sin any more than one can separate ... a bone from the mouth of a very hungry dog!

Either one 'cancels' the whole sin(x+2) or nothing at all.
 
  • #3
You don't cancel the (x+2)'s. You should know

[tex]\lim_{x\rightarrow 0}\frac{\sin x}{x}[/tex]

Use that somehow.
 
  • #4
Did you mention in class that lim(a x b) = lim(a) x lim(b)?
 
  • #5
Yes, we mentioned all the above, and this was much easier after LCKurtz's suggestion:

=lim sin((x+2)/(x+2)) * lim (x-1)/(x-2)
= 1 * -3/-4
=3/4
I really need to learn how to use that equation generator thing...

Much appreciated folks! :)
 

What is the definition of the limit of a sin function?

The limit of a sin function is the value that the function approaches as the input approaches a specific value. In simpler terms, it is the y-value that the function gets closer and closer to as the x-value gets closer and closer to a specific value.

How do you find the limit of a sin function?

To find the limit of a sin function, you can use the graphical method by plotting the function and observing the behavior of the y-values as the x-values approach the specific value. You can also use algebraic methods such as substitution or factoring to simplify the function and evaluate the limit.

What is the difference between a one-sided limit and a two-sided limit for a sin function?

A one-sided limit for a sin function only considers the behavior of the function as the x-value approaches the specific value from one direction (either from the left or right). A two-sided limit, on the other hand, considers the behavior of the function as the x-value approaches the specific value from both directions.

When does the limit of a sin function not exist?

The limit of a sin function does not exist if the function has a vertical asymptote at the specific value or if the function oscillates and does not approach a particular value as the x-value gets closer to the specific value.

Why is the limit of a sin function important in mathematics and science?

The limit of a sin function is important because it helps us understand the behavior of the function at a specific point. It is also used in various applications, such as calculating the maximum and minimum values of a function, determining the continuity of a function, and finding the derivatives of trigonometric functions.

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