Limits question involving trigonometric functions

In summary, the given conversation discusses the function f(x) and its limit as x approaches infinity. The limit is found to be 1. The next part asks to show the existence of a number c in the interval (0,1) where f(c)=1. The conversation also mentions finding the maximum and minimum values of (cos(x))^2, but does not provide a solution. Finally, the values of f(0) and f(1) are not mentioned.
  • #1
dustinm
7
0
f(x)= ((cosx)[itex]^{2}[/itex]+1)/e[itex]^{x}[/itex][itex]^{2}[/itex]

So for the limit of f(x) as x→∞ I would just input ∞ for x. I'm confused after this though, wouldn't it just be ∞/∞ = 1?

the next part says show that there exists a number c ε (0,1) that f(c)=1
I don't know what this is asking for me to solve.
 
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  • #2
dustinm said:
f(x)= ((cosx)[itex]^{2}[/itex]+1)/e[itex]^{x}[/itex][itex]^{2}[/itex]

So for the limit of f(x) as x→∞ I would just input ∞ for x. I'm confused after this though, wouldn't it just be ∞/∞ = 1?
What is the maximum value of (cos(x))2? the minimum value of (cos(x))2?

the next part says show that there exists a number c ε (0,1) that f(c)=1
I don't know what this is asking for me to solve.
What is f(0) ?

What is f(1)?
 

Related to Limits question involving trigonometric functions

1. What is a limit involving a trigonometric function?

A limit involving a trigonometric function is a mathematical concept that describes the behavior of a function as the input approaches a certain value. It is often used to determine the value of a function at a point that is not defined, or to study the behavior of a function near a certain point.

2. How do you find the limit of a trigonometric function?

To find the limit of a trigonometric function, you can use algebraic manipulation, trigonometric identities, and the properties of limits. You can also use L'Hopital's rule for certain types of limits involving trigonometric functions.

3. What are some common trigonometric limits?

Some common trigonometric limits include:
1. The limit of sin(x)/x as x approaches 0 is equal to 1
2. The limit of (1-cos(x))/x as x approaches 0 is equal to 0
3. The limit of (1-cos(x))/x^2 as x approaches 0 is equal to 1/2
4. The limit of tan(x)/x as x approaches 0 is equal to 1
5. The limit of sin(ax)/sin(bx) as x approaches 0 is equal to a/b if a and b are non-zero constants

4. Why are limits involving trigonometric functions important?

Limits involving trigonometric functions are important because they help us understand the behavior of these functions and their graphs. They also have real-life applications in fields such as physics, engineering, and economics.

5. How do you solve a limit involving a trigonometric function at infinity?

To solve a limit involving a trigonometric function at infinity, you can use the following rules:
1. If the limit contains a rational function, the limit is equal to the ratio of the leading coefficients of the numerator and denominator
2. If the limit contains a polynomial function, the limit is equal to the leading term of the polynomial
3. If the limit contains a trigonometric function, the limit is equal to the leading term of the trigonometric function
4. If the limit contains an exponential function, the limit is equal to 0 or infinity depending on the sign of the exponent
5. If the limit contains a logarithmic function, the limit is equal to -∞ or ∞ depending on the sign of the argument

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