Calculus Homework

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MAT 265 Written Homework #1
1.3 – 1.5, 2.2 Due: September 10
Solve the following problems, showing any necessary work. You must give EXACT answers, not decimal
approximations (unless stated otherwise).
1. Find the following limits algebraically.
a. [1 point] limx→5
3x − 15

2 x + 6 −

4 x − 4
b. [1 point] limx→7
14
x

10
x − 2
1 −
35
x
2 − 2x
c. [1 point] lim
θ→π/4
(sin θ − 2 tan θ) (Hint: π
4
is a special angle.)
2. [3 points] Let f be the function defined by
f(x) =



2x
2 − 3x if x < 0
0 if x = 0
−3x
2 − x if (0 < x) and (x < 2)
−18 if x = 2
−3x
2 − 2 if (2 < x) and (x < 3)
−29 if x = 3
−2x
2 − 3x − 1 if 3 < x
For each of a = 0, a = 2, and a = 3, find lim
x→a−
f(x), lim
x→a+
f(x), and limx→a
f(x), (writing DNE if that
limit does not exist), and determine whether f(x) is continuous at x = a.
3. [2 points] Use the Intermediate Value Theorem to show that there is a solution to the equation
e
x
sin x − x
3 = 2021
such that 7 ≤ x ≤ 8. (You do not need to solve this equation.)
4. [1 point] Use the limit definition of the derivative to find f
0
(x), where f(x) = 3x
2 + 7x − 5. [Moved to
Homework #2]

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