You are missing parentheses. Is this "cos(1)- (sin(t)/t)" or is it "cos(1- (sin(t)/t))"?odmart01 said:Homework Statement
lim t--> 0 cos(1-(sint/t)
Homework Equations
lim theta-->0 sin(theta)/theta =1
The Attempt at a Solution
I usually don't have a problem with these limits, but I've never done 1 with a trig function inside another trig function. So I don't know how to begin this one.
HallsofIvy said:You are missing parentheses. Is this "cos(1)- (sin(t)/t)" or is it "cos(1- (sin(t)/t))"?
If it is the first, then cos(1) is just a constant: the limit is cos(1)- 1.
If it is the second, then use the fact that cosine is continuous: the limit is cos(1- 1)= cos(0)= 1.
odmart01 said:it is cos(1- (sin(t)/t)), but what do you do with the (sin(t)/t) in the inside, you can't just assume its 0 because then it does not exist. How are you getting 1-1. Can you explain?
Inferior89 said:It exists.
You said yourself in the "relevant equations" part that:
lim theta-->0 sin(theta)/theta =1
which is the same as
lim t -->0 sin(t)/t = 1
The limit of a trigonometric function is the value that the function approaches as the input approaches a certain value or point. This value may or may not be equal to the actual output of the function at that point.
The limit of a trigonometric function can be determined by evaluating the function at values very close to the given point and observing if the outputs approach a particular value. Alternatively, mathematical techniques such as L'Hôpital's rule or trigonometric identities can also be used to determine the limit.
The six common trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. Their corresponding limits are as follows:
- The limit of sine function as x approaches 0 is 0.
- The limit of cosine function as x approaches 0 is 1.
- The limit of tangent function as x approaches 0 is 0.
- The limit of cotangent function as x approaches 0 is undefined.
- The limit of secant function as x approaches 0 is 1.
- The limit of cosecant function as x approaches 0 is undefined.
Finding the limit of a trigonometric function is important in understanding the behavior of the function near a particular point. It can also be useful in solving real-world problems that involve trigonometric functions, such as finding maximum and minimum values or determining the behavior of a system over time.
Yes, the limit of a trigonometric function can be undefined if the function does not approach a particular value or if the function is discontinuous at that point. This can happen, for example, when the denominator of a trigonometric function becomes 0, resulting in an undefined value.