What is the Limit of a Function as x Approaches 0?

  • Thread starter bumpercars21
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In summary, to solve this limit problem, one should try using l'Hopital's rule and the Taylor series expansions of cosine and sine to simplify the expression. By subtracting and factoring out x terms, the limit can then be calculated.
  • #1
bumpercars21
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Homework Statement



http://i45.tinypic.com/20rsnis.jpg

Homework Equations



Compute the lim

The Attempt at a Solution


I tried using l'Hopital's rule but I have to keep finding the derivative and it doesn't yield an answer. I have to use taylor series for the trig functions, but don't know how this will work
 
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  • #2
Go ahead, try it. Expand cos(2x) and sin(x) in a taylor series. See which terms cancel. If you don't know the expansion of cos and sin, they are easy to look up.
 
  • #3
http://en.wikipedia.org/wiki/Taylor_series

a good rule of thumb is to turn sin and cosine and even e into their taylor series whenever doing limit problems. then subtract whatever you can. and factor out as many x terms as you need to make the top and bottom non zero. and then calculate the limit.
 

FAQ: What is the Limit of a Function as x Approaches 0?

What does "lim as x approaches 0" mean?

"Lim as x approaches 0" refers to the limit of a function as the input value, x, gets closer and closer to 0. It is used to determine the behavior of a function near a specific point.

Why is it important to compute the limit as x approaches 0?

Computing the limit as x approaches 0 allows us to understand the behavior of a function at a critical point. It can help us determine if the function is continuous, has a vertical asymptote, or has a removable discontinuity at that point.

How do you compute the limit as x approaches 0?

To compute the limit as x approaches 0, you can either use algebraic manipulation or graphing to evaluate the function at values closer and closer to 0. Alternatively, you can use L'Hopital's rule, which states that the limit of the quotient of two functions is equal to the limit of their derivatives.

What if the limit as x approaches 0 is undefined?

If the limit as x approaches 0 is undefined, it means that the function does not have a defined value at that point. This could be due to a vertical asymptote or a removable discontinuity. Further analysis is needed to determine the behavior of the function at that point.

Can we always compute the limit as x approaches 0?

In some cases, the limit as x approaches 0 may not exist. This could be due to the function approaching different values from the left and right sides of 0, or oscillating infinitely. In such cases, the limit is said to be undefined.

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