The limit of lim x approaches 0 of sin(x)/x is 1. This fundamental limit is often encountered in calculus and is important for understanding the behavior of the sine function near zero.
To explain why this is the case, we can use two methods: the squeeze theorem and L’Hôpital’s rule.
1. **Squeeze Theorem**: As x approaches 0, the sine function behaves similarly to the x value itself. More formally, it can be shown that:
sin(x) < x < tan(x)
Dividing through by x (which is positive as we approach 0 from the right), we get:
sin(x)/x < 1 < (tan(x))/x
As x approaches 0, both sin(x)/x and tan(x)/x squeeze towards 1. Thus, by the squeeze theorem, we find that lim x approaches 0 of sin(x)/x equals 1.
2. **L’Hôpital’s Rule**: Since the limit produces a 0/0 indeterminate form, we can apply L’Hôpital’s rule, which involves taking the derivative of the numerator and denominator:
Taking the derivative of the numerator:
d/dx[sin(x)] = cos(x)
And the derivative of the denominator:
d/dx[x] = 1
Applying L’Hôpital’s rule gives us:
lim x approaches 0 of sin(x)/x = lim x approaches 0 of cos(x)/1 = cos(0) = 1
In conclusion, both methods lead us to the same result: the limit of sin(x)/x as x approaches 0 is indeed 1.