Let's do some calculus! (13)

Calculus Level 3

$f(x) = \begin{cases} \dfrac{\sin \left \lfloor x \right \rfloor}{\left \lfloor x \right \rfloor}, & \text{for} & \left \lfloor x \right \rfloor \neq 0 \\ \\ 0, & \text{for} & \left \lfloor x \right \rfloor = 0 \end{cases}$

If $f(x)$ is as defined above, find $\displaystyle \lim_{x \to 0^{-}}{f(x)}$ .

Notation: $\lfloor \cdot \rfloor$ denotes the floor function .

For more problems on calculus, click here .

\sin 1

0

1

\sin 1^{\circ}

1 solution

Chew-Seong Cheong
Sep 21, 2016

$\begin{aligned} \lim_{x \to 0^-} f(x) & = \lim_{x \to 0^-} \frac {\sin \lfloor x \rfloor}{\lfloor x \rfloor} & \small \color{#3D99F6}{\text{Note that }\lim_{x \to 0^-} \lfloor x \rfloor = -1 } \\ & = \frac {\sin (-1)}{-1} = \frac {- \sin 1}{-1} = \boxed{\sin 1} \end{aligned}$

Here you need to clarify the floor function. It is commonly used in Brilliant as floor function and not GIF. The link will clarify that.

Chew-Seong Cheong - 4 years, 8 months ago

Okay. Thanks. :)

Tapas Mazumdar - 4 years, 8 months ago

Let's do some calculus! (13)

For more problems on calculus, click here .

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