Name: Hazel Ann Carillo
Section Code: T144 / BSIT 2A
Problem Set 2
Find the limit of the following:
1. li
m
𝐱𝟐−𝟓𝐱+𝟔
𝑥→ 𝐱−𝟐
2
(x + 2)(x - 2)
x-2
lim
x→2
(x - 2)(x - 3) 2 – 3 = -1
x-2
2. lim 𝐱+𝟏
𝑥→−1 𝐱𝟑+𝐱𝟐+𝐱+𝟏
x+1
(x + 1)(x2 + 1)
1 1__
1
(-1)2 + 1 1+1 2
lim
3. √𝒙+𝟒 −𝟐
𝒙→𝟎 𝒙
x= √
x + 4+2
x+ 4+2
1
√ 0+4 +2
1 1
__ __
2+2 4
4. Graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits.
lim f(x) where f(x) =
x -1, x≤2
𝑥→2
2x – 3, x>2
ANSWER:
LEFT-HAND LIMIT
lim f(x) = lim(x - 1) = 2 - 1 = 1
x→2− x→2−
RIGHT-HAND LIMIT
lim f(x) = lim(2x − 3) = 2(2) – 3 = 1
x→2+ x→2+
both one-sided limits equal 111, so the two-sided limit exists and
lim f (x )=1
x →2
2
1
0
-1 1 2 3
-2
-3
5. A union contract guarantees an 8% salary increase yearly for 3 years. For a current salary of $30,000, the
salaries f(t) (in thousands of dollars) for the next 3 years are given by
30,000, 0<t≤1
f(t) = 32,400, 1<t≤2 where t represents the time in years. Show that the limit of f as t →2
34,992, 2<t≤3 does not exist.
ANSWER:
For values of t just less than 2, f(t)=32,400
lim f(t)=32,400
t→2−
for values of t just greater than 2, f(t)=34,992
since 32,400 ≠ 34,992, the left-hand and right-hand limits are not equal.
lim f(t) does not exist
t→2−
𝑓(𝑥+ℎ)−𝑓(𝑥)
6. Find lim . Given the function f(x) = x2 − 3x.
ℎ→0 ℎ
ANSWER:
f(x + h) = (x + h )2 - 3(x + h) = x2+ 2xh +h2 - 3x - 3h
f(x + h) - f(x) = (x2 + 2xh + h2 - 3x - 3h) - (x2 − 3x) = 2xh + h2 - 3h
f(x + h) - f(x) 2xh + h2 – 3h
2x + h - 3
h h
lim(2x+h−3) = 2x + h - 3
h→0
