Is the Limit of Sin x/x=1 Proven in Elementary Calculus?

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In summary, the conversation discusses the equivalence of two limits, (lim x-->0) ((sin x)/x) = 1 and (lim x-->0) sin x = x, and whether the latter can be used to prove the former. It is concluded that while the two limits are equal, it is not a sufficient condition for the limit (lim x-->0) ((sin x)/x) = 1 to exist.
  • #1
agapito
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From elementary calculus it is known that

(lim x-->0) ((sin x)/x) = 1.

Is this result equivalent to (lim x-->0) sin x = x ?

If so, how is it proved? Many thanks for all guidance.
 
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  • #2
No, that makes no sense.
The left is a number, the limit of sin(x) as x goes to 0 (which happens to be 0) while the right is a function, x.

What is true is that $\lim_{x\to 0} sin(x)= \lim_{x\to 0} x$ which is simply 0= 0.
 
  • #3
OK thanks for responding
 
  • #4
I think I should point out that while $\lim_{x\to a} f(x)= \lim_{x\to a}g(x)$ is a necessary condition for $\lim_{x\to a}\frac{f(x)}{g(x)}= 1$ it is not sufficient.

For example $\lim_{x\to 0} sin(x)= \lim_{x\to 0} x^2= 0$ but $\lim_{x\to 0}\frac{sin(x)}{x^2}$ does not exist.
 

FAQ: Is the Limit of Sin x/x=1 Proven in Elementary Calculus?

What is the limit of Sin x/x as x approaches 0?

The limit of Sin x/x as x approaches 0 is equal to 1. This can be proven using the squeeze theorem or by evaluating the limit using trigonometric identities.

How is the limit of Sin x/x related to the concept of a derivative?

The limit of Sin x/x is equivalent to the derivative of Sin x at x=0. This is because the derivative of Sin x is equal to Cos x, and when x=0, Cos x=1.

Can the limit of Sin x/x be solved using L'Hopital's rule?

Yes, L'Hopital's rule can be used to solve the limit of Sin x/x. By taking the derivative of both the numerator and denominator, the limit can be rewritten as Cos x/1, which evaluates to 1 when x=0.

Is the limit of Sin x/x always equal to 1?

Yes, the limit of Sin x/x is always equal to 1. This can be verified by graphing the function or by evaluating the limit at different values of x.

Can the limit of Sin x/x be extended to other trigonometric functions?

Yes, the limit of Sin x/x can be extended to other trigonometric functions such as Cos x/x and Tan x/x. The limit of Cos x/x is also equal to 1, while the limit of Tan x/x is equal to 0.

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