Question

Limx→0f\left(x\right)\:=\frac{\sin \left(0\right)}{0}

Answer 1

To evaluate the limit of the function as x approaches 0, where f(x) = sin(x)/x, we can use L'Hôpital's rule.

According to L'Hôpital's rule, if we have an indeterminate form of the type 0/0 or ∞/∞, we can take the derivative of the numerator and denominator until we reach a determinate form.

Let's differentiate the numerator and denominator:

`\begin{align}f(x) &= (\sin(x))/(x) \\f'(x)&=(d)/(dx)(\sin(x)) \\ &= \cos(x)\end{align}`

`\begin{align}f(x) &= (\sin(x))/(x) \\f'(x)&=(d)/(dx)(x) \\ &= 1\end{align}`

Now we can evaluate the limit as x approaches 0 using the derivatives:

`\begin{align}\lim_(x \to 0) f(x) &= \lim_(x \to 0) (\sin(x))/(x) \\ &= \lim_(x \to 0) (\cos(x))/(1) \quad \text{(using L'Hôpital's rule)} \\ &= \cos(0) \\ &= 1 \end{align}`

Therefore, the limit of f(x) as x approaches 0 is 1.