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Find the value of the given Lim x tending to 0 cos(tanx) - cosx/x^4

Himanshu yadav , 5 Years ago
Grade 12
anser 1 Answers
Samyak Jain

Last Activity: 5 Years ago

Let L be the given limit.
\therefore L = lim x\rightarrow0 [cos(tanx) – cosx] / x4      ...Using cosA – cosB = 2 sin{(A + B)/2} sin{(A – B)/2}
       = lim x\rightarrow0 [2 sin{(tanx + x)/2} sin{(x – tanx)/2}] / x4
Multiply and divide by {(tanx + x)/2} {(x – tanx)/2}
  L   = 2 . lim x\rightarrow0 sin{(tanx + x)/2} . {(tanx + x)/2} / (tanx + x)/2 . 
                    sin{(x – tanx)/2} . {(x – tanx)/2} / {(x – tanx)/2} / x4
 
As x\rightarrow0, (tanx + x)/2 \rightarrow0 and (x – tanx)/2 \rightarrow0.
L = 2 . lim x\rightarrow0 sin{(tanx + x)/2} / (tanx + x)/2 .lim x\rightarrow0 sin{(x – tanx)/2} / {(x – tanx)/2} . 
        lim x\rightarrow0 {(tanx + x)/2} . {(x – tanx)/2} / x4
 
We know that lim x\rightarrow0  tanx / x = 1
\therefore lim x\rightarrow0 sin{(tanx + x)/2} / (tanx + x)/2 = 1
and lim x\rightarrow0 sin{(x – tanx)/2} / {(x – tanx)/2} = 1
 
So, L = 2 . 1 . 1 . lim x\rightarrow0 {(tanx + x) / 2} . {(x – tanx) / 2} / x4
          = 2 lim x\rightarrow0 (tanx + x) . (x – tanx) / 4x4
          = ½ lim x\rightarrow0 (x2 – tan2x) / x4
Here use expansion of tangent series.
tanx = x + (1/3) x3 + (2/15) x5 + ….
tan2x = x2 + (2/3) x4 + f(x), where degree of x in f(x) is greater than 4.
\therefore L = ½ lim x\rightarrow0 [ x2 – {x2 + (2/3) x4 + f(x)} ] / x4
L = ½ lim x\rightarrow0 [ x2 – x2 – (2/3)x4 –  f(x)} ] / x4
L = ½ lim x\rightarrow0 [– (2/3) x4 –  f(x)} ] / x4
L = ½ lim x\rightarrow0 [–(2/3) x4 / x4 ] – ½ lim x\rightarrow0 [ f(x) / x4 ] = (1/2) (– 2/3)  =  – 1/3
\because f(x) contains powers of x greater than 4,  lim x\rightarrow0 [ f(x)} ] / x4 = 0,
\therefore lim x\rightarrow0 [cos(tanx) – cosx] / x4 = – 1/3.

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