What is the limit of sin(x)/x as x approaches 0?

To find the limit of $\lim x\to 0\left(\sin \right(x)/x)$, we start by substituting $x=0$. This gives us:

 

$\sin \left(0\right)/0=0/0$, which is an indeterminate form. Therefore, we need to use a different approach to evaluate this limit.

 

One common method is using L’Hôpital’s Rule, which states that if we have an indeterminate form of type 0/0 or ∞/∞, we can take the derivative of the numerator and the derivative of the denominator:

 

$\lim x\to \left(\sin \right(x)/x)$ = $\lim x\to \left(\cos \right(x)/1)$

 

Now we can substitute $x=0$ again:

 

$\cos \left(0\right)/1=1$.

 

Thus, the limit we are looking for is:

 

.

 

Another way to understand this limit intuitively is through the Squeeze Theorem. If we analyze the behavior of the sine function, we can observe that:

 

• For small values of x, $\sin \left(x\right)ext\{is\; approximately\; equal\; to\}$ x$.$
• Thus, as x approaches 0, the quotient approaches 1.

 

In conclusion, we find that:

 

lim_x→0 (sin(x)/x) = 1