Problem Set - 2
AUTUMN 2016
MATHEMATICS-I (MA10001)
1. Determine the following limits:
ex − e−x − x
;x→0
(a)
x2 sin x
2
July 25, 2016
(b) (2x tan x − π sec x); x → π/2
2
(c) cos(ax)b/x ; x → 0
(d) (sin x)tan x ; x → π/2
(e) (1 − x2 )1/ ln(1−x) ; x → 1
(f)
asin x − a
; x → π/2
ln sin x
(g) (sec x)cot x ; x → π/2
(h)
1 − 4 sin2 (πx/6)
;x→1
1 − x2
(1 + x)1/x − e
(i)
;x→0
x
(j)
tan x
1
(k)
;x→0
x
(l)
logsec( x2 ) cos x
logsec x cos x2
;x→0
x2 sin(1/x)
;x→0
tan x
2. Using L’Hospital rule evaluate
2
(a) lim (ln cot x)
tan x
x→0
sinx
(d) lim √
x→0
1 − cosx
1
(c) lim x ln(ex )−1
x→0
x
1
−
]
x→1 x − 1
ln x
tan x 1
(g) lim
x→0
x
(e) lim [
(i)
π
lim (cos x) 2 −x
x→π/2
e1/x − 1
(b) lim
x→∞ 2 arctan x2 − π
1
(f) lim (cot x) ln x
x→0
a x
(h) lim 1 +
x→∞
x
(j) lim (tan
x→1
πx tan πx
) 2
4
3. Expand in power of x − 2 of the polynomial x4 − 5x3 + 5x2 + x + 2.
4. Expand in power of x + 1 of the polynomial x5 + 2x4 − x2 + x + 1.
√
5. Write Taylor’s formula for the function y = x when a = 1, n = 3.
MA10001
DEPARTMENT OF MATHEMATICS
PROBLEM SET - 2
√
6. Write the Maclaurin formula for the function
y = 1 + x when n = 2. Further, estimate
√
the error of the approximate equation 1 + x ≈ 1 + 12 x − 81 x2 when x = 0.2.
7. Write down the Taylor’s expansion for the function f (x) = sinx about the point a =
n = 4.
π
4
with
8. If f 00 (x) exists on [a, b] and f 0 (a) = f 0 (b) prove that
1
a+b
= [f (a) + f (b)] + (b − a)2 f 00 (c) for some c ∈ (a, b).
f
2
2
2
9. Applying Taylor’s theorem with remainder prove that 1 + x2 − x8 <
√
1 + x < 1 + x2 if x > 0.
10. Applying Maclaurin’s theorem with the remainder to expand
(a) ln(1 + x)
(b) (1 + x)m
11. Using Taylor’s formula, evaluate
x − sin x
(a) lim
2
x→0 ex − 1 − x − x
2
1
(c) lim [x − x2 ln(1 + )]
x→0
x
2(tan x − sin x) − x3
x→0
x5
(b) lim
(d) lim (
x→0
1
cot x
−
)
2
x
x
Note: Submit solution of problems 1,3,4,7 to your tutorial teacher next week.
2