Is L’Hopital a good idea in this case?

Find aa and bb so that function ff is continuous.

f(x)={sin(3x)(eax−1)2×3:x<0ax2+3x+b:0≤x<1tan2(x−1)(x−1)2:x≥1\begin{eqnarray*} f(x)=\left\{\begin{array}{lcc} \frac{\sin(3x)(e^{ax}-1)}{2x^3} &:&x<0 \\ ax^2+3x+b &:& 0\le x <1\\ \frac{\tan^2(x-1)}{(x-1)^2} &:&x\ge1 \end{array} \right. \end{eqnarray*} Well, I've been thinking on this and no way except L'Hأ´pital's rule has come to my mind, and this way takes a while to solve. Is there any better way? =================      Did you think about Taylor series ? – Claude Leibovici 2 days ago      @ClaudeLeibovici Not yet – Ali Seyfi 2 days ago ================= 2 Answers 2 ================= Hints: limx→0−sin(3x)xeax−1x \lim_{x\to0^-}\frac{\sin(3x)}{x}\frac{e^{ax}-1}{x} is finite. But you still have another xx at the denominator, so the limit is infinite unless… limx→1+tan(x−1)x−1=limt→0+tantt=… \lim_{x\to1^+}\frac{\tan(x-1)}{x-1}= \lim_{t\to0^+}\frac{\tan t}{t}=…      I would say "unless.... a=0\;a=0\; ", but then also \;b=0\;\;b=0\; and ...etc . – DonAntonio 2 days ago      a must be 0, that's ok but why b=0? – Ali Seyfi 2 days ago      tan part's limit will be 1 so lim ax^2+3x+b lim ax^2+3x+b when x-->1 from left should be 1 too.
– Ali Seyfi
2 days ago

  

 

@AliSeyfi What’s f(0)f(0)?
– egreg
2 days ago

  

 

@egreg got it. tnx
– Ali Seyfi
2 days ago

Hints:
Knowing
\lim_{x\to0}\frac{\sin x}{x}=\lim_{x\to0}\cos x\frac{\tan x}{x}=1,\lim_{x\to0}\frac{\sin x}{x}=\lim_{x\to0}\cos x\frac{\tan x}{x}=1,

\lim_{x\to0}\frac{e^x-1}x=1,\lim_{x\to0}\frac{e^x-1}x=1,

L’Hospital is unnecessary.