Can someone help explain how to solve this limit by taking the natural log of bo

Question

Can someone help explain how to solve this limit by taking the natural log of both sides? I've included my steps taken below if that helps. The actual answer is 1/e^(2/5) but I keep getting the reciprocal "e^(2/5)"

1: y=(10x/(10x+1))^4x

2: ln(y)=ln(10x/(10x+1))^4x

3: ln(y)=4x(ln(10x/(10x+1)))

4: ln(y)=4x(ln(10x)-ln(10x-1))

5: ln(y)=(4(ln(10x)-ln(10x-1))(x^-1)

Now I differentiate top and bottom in accordance to L'Hopital"s rule and simplify to get:

6: ln(y)=4x/(10x-1)

plugging in infinity gives:

7: ln(y)=2/5

8: e^ln(y)=e^(2/5)

9: y=e^(2/5)

Would greatly appreciate someone telling me what I'm doing wrong. I think I'm on the right track since I keep getting the reciprocal of the answer.

Thanks

Explanation / Answer

In step 4, there have to be -ln(10x+1) that is

ln(y)=4x(ln(10x)-ln(10x+1))

Step5 . ln(y)=(4(ln(10x)-ln(10x+1))(x^-1)

On differentiating

Step 6. ln(y) = -4x/(10x+1)

Plugging infinity gives

Step 7. ln(y) = -2/5

Step 8 . y= e-2/5= 1/e^(2/5)

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