limx0esinxesinx2tanxtanxx=?

The given limit,

=limx0(1+sinx+sin2x2!+sin3x3!+.)(1sinx+sin2x2!sin3x3!+.)2(x+x33+215x5+.)(x+x33+215x5+.)x

=limx02(sinx+sin3x3!+sin5x5!+.)2(x+x33+215x5+.)x33+215x5+.

=limx02(sinxx+sin3x3!x33+sin5x5!215x5+.)x33+215x5+.

=limx02(xx33!+x55!..x+sin3x3!x33+sin5x5!215x5+.)x33+215x5+.

=limx02(x33!+x55!..+sin3x3!x33+sin5x5!215x5+.)x33+215x5+.

=limx02(13!+x25!..+sin3xx33!13+sin5xx35!215x2+.)13+215x2+.

=2(13!+13!13)13=2

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