Limit with trigonometric functions

In summary, the limit as x approaches pi from the left of Tanx/(1-Cosx) does not equal the limit as x approaches pi from the left of Sec2x/Sinx, both of which equal infinity. The first approach of plugging pi into x gives a result of 0/2, which is not an indeterminate form and therefore does not require the use of L'Hopital's Rule. However, the original limit is not in either of the indeterminate forms, so L'Hopital's Rule cannot be applied. The simplification of 0/2 to just 0 is not accurate and the correct answer is false.
  • #1
Jules18
102
0

Homework Statement



True or false?

lim Tanx/(1-Cosx) = lim Sec2x/Sinx = Infinity

(limits are as x approaches [tex]\pi[/tex] from the left)

The Attempt at a Solution



I tried just plugging [tex]\pi[/tex] into x in the first limit, and I ended up getting 0/2, which exists but is just 0. So I said false. Am I simplifying it too much? because what I did seems to make sense but it looks like that's not what they wanted me to do.
 
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  • #2


Isn't it just 0/2=0?
 
  • #3


The original limit was neither of the indeterminate forms [0/0] or [infinity/infinity], so L'Hopital's Rule doesn't apply here. Your first approach was correct.
 

FAQ: Limit with trigonometric functions

1. What is a limit with trigonometric functions?

A limit with trigonometric functions is a mathematical concept that describes the behavior of a function as its input values approach a certain value. It is typically used to determine the value of a function at a point where it is undefined or to analyze the behavior of a function near a certain point.

2. How do you calculate a limit with trigonometric functions?

To calculate a limit with trigonometric functions, you can use algebraic manipulation, trigonometric identities, and the properties of limits. You may also need to use L'Hospital's rule or evaluate the function at different values approaching the limit.

3. What are the common trigonometric functions used in limits?

The most commonly used trigonometric functions in limits are sine, cosine, tangent, cotangent, secant, and cosecant. These functions are used because they have well-defined limits and are useful in analyzing the behavior of a function near a certain point.

4. How do you solve indeterminate forms involving trigonometric functions?

To solve indeterminate forms involving trigonometric functions, you can use trigonometric identities to rewrite the expression in a form that is easier to evaluate. You may also need to use L'Hospital's rule to take the derivative of the numerator and denominator separately.

5. What are the real-life applications of limits with trigonometric functions?

Limits with trigonometric functions are used in a variety of real-life applications, including physics, engineering, and economics. For example, they can be used to model the motion of a pendulum, analyze the stability of a bridge, or predict the growth of a population. They are also used in signal processing to filter out noise from a signal.

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