Calculus 1: Solving Limits Without L'Hopital

In summary, the conversation discusses a limit problem that made the person mad. They tried different methods to solve it but ultimately realized that the limit does not exist. They also mention that using L'Hopital's rule is not possible in this case. The conversation ends with the person being upset because the problem was a quiz question with a misleading answer.
  • #1
rocomath
1,755
1
So I was helping this person the other day, and this problem made me pretty mad.

[tex]\lim_{x\rightarrow\frac{\pi}{9}}\left(\frac{\sin x}{x-\frac{\pi}{9}}\right)[/tex]

First, I added and subtracted pi/9. I regrouped the angle so that it was sin[(x-pi/9)+pi/9] and expanded it, but that didn't help at all.

I could use L'Hopital, but this is a calculus 1 problem.
 
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  • #2
Might have made you mad because the limit doesn't exist. sin(pi/9) is not equal to zero. The denominator does approach zero.
 
  • #3
Yeah, like Dick already pointed it out, this limit doesn't exist, and it is very easy to notice it by just observing some values to the left and to the right of [tex] \frac{\pi}{9}[/tex] since pi/9 lies in the first quadrant, sin(x) won't change sing.

But the bottom, when we take values to the left of pi/9 will be negative, while to the right positive.
SO the right hand side limit of this is infinity, while the left hand side -infinity. The overall limit does not exist.
 
  • #4
And, therefore, you could not use L'Hopital's rule!
 
  • #5
HallsofIvy said:
And, therefore, you could not use L'Hopital's rule!
oh shoot! This was a quiz problem and gave the answer -1. Now I'm mad!
 

Related to Calculus 1: Solving Limits Without L'Hopital

1. What is a limit in calculus?

A limit in calculus is a fundamental concept that describes the behavior of a function as the input approaches a certain value. It represents the value that the function is approaching, rather than the actual value at that point. Limits are essential in calculus because they allow us to understand and analyze the behavior of functions, particularly at points where they may be undefined or have discontinuities.

2. How do you solve limits without L'Hopital's rule?

To solve limits without L'Hopital's rule, you can use algebraic manipulation, special limits, and trigonometric identities. For algebraic manipulation, you can factor, rationalize, or simplify the expression to eliminate any indeterminate forms. For special limits, you can use known limit values or theorems to evaluate the limit. Trigonometric identities can also be used to simplify trigonometric expressions and evaluate the limit.

3. What are the common types of limits encountered in Calculus 1?

The common types of limits encountered in Calculus 1 are polynomial, rational, exponential, logarithmic, and trigonometric limits. Polynomial limits involve functions with powers of x, while rational limits involve fractions with polynomial functions in the numerator and denominator. Exponential limits involve functions with a base of e, and logarithmic limits involve functions with a base of 10 or e. Trigonometric limits involve functions such as sine, cosine, and tangent.

4. How do you determine if a limit exists?

A limit exists if the left-hand limit (limit as x approaches the value from the left) and the right-hand limit (limit as x approaches the value from the right) are equal. If the left-hand limit and the right-hand limit both approach the same value, then the limit exists. Additionally, the limit does not exist if the two-sided limits are different, or if one or both of the one-sided limits are undefined or infinite.

5. When is L'Hopital's rule used in solving limits?

L'Hopital's rule is used in solving limits when you have an indeterminate form of 0/0 or ∞/∞. This rule states that if you have a limit of the form f(x)/g(x), where both f(x) and g(x) approach 0 or ∞ as x approaches a certain value, then the limit can be evaluated by taking the derivative of both f(x) and g(x) and then finding the limit of the resulting quotient. However, L'Hopital's rule should only be used when other methods, such as algebraic manipulation and special limits, are not applicable.

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