dear student, arun is just making excuses. trust me the pic is clearly visible. he cn also download it for better visibility.
coming to ur ques, such ques are best done using series expansion.
f(x)= x(x^1/1! – x^3/3! + x^5/5! – …...)/2 – 1 + 1 – x^2/2! + x^4/4! – ….....
= x^2/2 – x^4/12 + …... – x^2/2! + x^4/24 – …......
= – x^4/12 + x^6/240 – ….. + x^4/24 – …
= x^4( – 1/12+x^2/240 – …. + 1/24 – …...)
hence, Lt f(x)/x^k= Lt [x^4/x^k]*( – 1/12+x^2/240 – …. + 1/24 – …...)
= Lt [x^4/x^k]* Lt ( – 1/12+x^2/240 – …. + 1/24 – …...)
= Lt x^(4 – k) * ( – 1/12 + 1/24)
now, Lt x^(4 – k) exists finitely and non zero only when k=4. if k is less than 4, then 4 – k is greater than zero and hence lim becomes 0. if k is greater than 4, then k – 4 is greater than zero and hence lim becomes Lt 1/x^(k – 4)= ± infinity
so, k=4
KINDLY APPROVE :))
