14.12 Reinforcement
- (i) Find the values of the function( 3 x+ 1 )/xwhenxhas the values 10, 100, 1000,
1,000,000.
(ii) What limit does the function approach asxbecomes infinite?
2.Find lim
x→ 2
x^2 − 4
x^2 − 2 x
3.Find lim
x→∞
4 x^2 +x− 1
3 x^2 + 2 x+ 1
4.The distance fallen by a particle from rest is given bys= 16 t^2 m. Representing an
increase in time byδtand the corresponding increase insbyδs, find an expression
forδsin terms ofδtand hence findδs/δt. Using this result find the average velocity
for the following intervals:
(i) 2secs.to2.2secs. (ii) 2secs.to2.1secs.
(iii) 2 secs. to 2.01 secs. (iv) 2 secs. to 2.001 secs.
From these results infer the velocity at the end of 2 secs.
5.Which of the following functions exist at the stated values ofx? If the functions do
not exist, find the limit of the function at the values concerned, if it exists.
(i) x^2 |x= 4 (ii)
(x− 1 )(x+ 2 )
x− 1
∣
∣
∣
x= 1
(iii)
(x+ 4 )^4
(x+ 4 )
∣
∣
∣
∣
x=− 4
(iv)
1
x
∣
∣
∣
x= 0
(v)
x^2 − 4
x+ 2
∣
∣
∣
x=− 2
(vi)
x^2 + 1
x^4 + 1
∣
∣
∣
x=− 1
(vii)
x^4 − 16
x− 2
∣
∣
∣
x= 2
(viii)
x+ 1
x− 1
∣
∣
∣
x= 1
6.Given lim
x→ 0
sinx
x
=1 find the following limits:
(i) lim
x→ 0
sin 2x
2 x
(ii) lim
x→ 0
sin 3x
x
(iii) lim
x→ 0
sinαx
βx
α
= 0
β
= 0
(iv) lim
x→ 0
(
sinx
x
) 3
(v) lim
x→ 0
(
sinαx
βx
)γ
(vi) lim
x→ 0
sin^2 x
x
(vii) lim
x→ 0
sinx
π−x
7.Which of the functions in Q5 are continuous for allx? Give any points of discontinuity
and where possible give a function which is continuous everywhere and is equal to
the given function away from the points of discontinuity.
8.Find the limit of
(x+h)^3 −x^3
h
ash→0. Hence find the slope of the curvey=x^3
at the pointx=1.
