Limits Plus: V

Calculus Level 3

Evaluate

lim ⁡ x → 0 sin ⁡ ( a + 2 x ) − 2 sin ⁡ ( a + x ) + sin ⁡ a x 2 \large \displaystyle \lim_{x\rightarrow 0}{\dfrac{\sin{(a+2x)}-2\sin{(a+x)}+\sin{a}}{x^2}}

without the application of L'Hôpital's Rule.

Problem credit: Calculus: 6E, James Stewart

− cos ⁡ a -\cos{a} − sin ⁡ a -\sin{a} sin ⁡ a \sin{a} − csc ⁡ a -\csc{a}

John M. by John M. , 24, USA

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1 solution

Rohan Shah
May 22, 2015

just separate 2sin(a +x) as sin(a +x) + sin( a+ x);
so you can apply s-s = 2cs;

then you will get........... 2sin(x/2) [ cos (a + 3x/2 ) - cos ( a + x/2) ]
apply c-c = -2ss;

so; 2sin[ x/2] X (-2) X sin(a + x) X sin (x/2)
Just apply limit and lim x tends to 0 (sinx)/x = 1;

hence;
-sina.

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