my calculus teacher marked this limit question wrong but I think my answer is correct can you check it

Question

I had an exam question:

lim⁡x→0sin⁡(2x)−2sin⁡xx3

I wrote:

Using LHopitals rule:
lim⁡x→02cos⁡(2x)−2cos⁡x3x2
lim⁡x→0−4sin⁡(2x)+2sin⁡x6x
lim⁡x→0−8cos⁡(2x)+2cos⁡x6
=−8(1)+2(1)6=−66=−1

My teacher marked it wrong and said answer is 0. But I applied LHopital 3 times since it was 0/0 each time. Where did I go wrong?

I think it might be because $\sin x \approx x - \frac{x^3}

David Kim · Answer

For limits, I would first classify the expression before applying any rule.

If direct substitution gives a number, you are done. If it gives 0/0 or ∞/∞, then algebra, series expansion, or L'Hopital's rule may be useful. If it gives something like 0⋅∞ or ∞−∞, rewrite it into a quotient first.

For example, near 0 the expansion
sin⁡x=x−x36+O(x5)
often gives the cleanest answer. It also explains why repeated L'Hopital works in some cases but c

My calculus teacher marked this limit question wrong but I think my answer is correct can you check it

1 Answer