C.B.S.E Board problem

Calculus Level 3

Evaluate

l i m x → 0 6 t a n x − 6 s i n x t a n x − s i n x lim_{x \to 0} \huge{\frac{6^{tanx} - 6^{sinx}}{tanx - sinx}}

Write the correct answer upto 3 decimals places


The answer is 1.791.

U Z by U Z , 24, Guyana

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1 solution

Aritra Jana
Nov 29, 2014

We have:

lim ⁡ x → 0 6 t a n x − 6 s i n x t a n x − s i n x \large{\lim\limits_{x\to 0}\dfrac{6^{tanx}-6^{sinx}}{tanx-sinx}}

we write 6 t a n x = 6 ( t a n x − s i n x ) + s i n x = 6 s i n x 6 t a n x − s i n x \large{6^{tanx}=6^{(tanx-sinx)+sinx}=6^{sinx}6^{tanx-sinx}}

∴ lim ⁡ x → 0 6 s i n x 6 t a n x − s i n x − 1 t a n x − s i n x \therefore \large{\lim\limits_{x\to 0}6^{sinx}\dfrac{6^{tanx-sinx}-1}{tanx-sinx}}

= lim ⁡ s i n x → 0 6 s i n x . lim ⁡ t a n x − s i n x → 0 6 t a n x − s i n x − 1 t a n x − s i n x =\large{\lim\limits_{sinx\to 0}6^{sinx}.\lim\limits_{tanx-sinx\to 0}\dfrac{6^{tanx-sinx}-1}{tanx-sinx}}

= 1. l o g e 6 =1.log_{e}6 ...(by standard limits)

= 1.791 =\boxed{1.791}

9 Helpful 0 Interesting 0 Brilliant 0 Confused

Yes that's why i posted the question. we get answer in one line . upvoted!

U Z - 6 years, 6 months ago

Nice solution ! I did the same. :)

Keshav Tiwari - 6 years, 6 months ago

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