F(x)=(36^x-9^x-4^x+1)/√2-√1+cos x for x not equal to 0 and K for x=0. Find value of K? Function is continuous at x=0 Thanks

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Answer 1 · 2017-06-28T17:35:08

# K=(4sqrt2)(ln4)(ln9).#

The function #F,# is defined by,

# F(x)=(36^x-9^x-4^x+1)/(sqrt2-sqrt(1+cosx)), if x ne0:#

#F(x)=K, if x=0.#

Knowing that, #1+cos2theta=2cos^2theta, and, #

# 1-cos2theta=2sin^2theta," we see that, for "xne0,#

#F(x)=(36^x-9^x-4^x+1)/(sqrt2-sqrt(1+cosx)),#

#={9^x(4^x-1)-1(4^x-1)}/{sqrt2-sqrt(2cos^2(x/2))},#

#={(4^x-1)(9^x-1)}/{(sqrt2)(1-cos(x/2))},#

#={(4^x-1)(9^x-1)}/{(sqrt2)(2sin^2(x/4))},#

#=1/(2sqrt2){(4^x-1)/x}{(9^x-1)/x}(x/sin(x/4))(x/sin(x/4)),#

#=1/(2sqrt2){(4^x-1)/x}{(9^x-1)/x}((x/4*4)/sin(x/4))((x/4*4)/sin(x/4)),#

#=16/(2sqrt2){(4^x-1)/x}{(9^x-1)/x}((x/4)/sin(x/4))((x/4)/sin(x/4)),#

Now, recall that, #lim_(h to 0)(a^h-1)/h=lna, and, lim_(h to 0)sinh/h=1.#

# :. lim_(x to 0)F(x)=(4sqrt2)(ln4)(ln9)(1)(1)=(4sqrt2)(ln4)(ln9).#

#because," F is continuous at "x=0, :., lim_(x to 0)F(x)=F(0)=K,#

# rArr K=(4sqrt2)(ln4)(ln9).#

Enjoy Maths.!