Mathematics 100 – Midterm Exam Review
INVERSE FUNCTIONS
1. Consider the function given by f x ln x3 1 .
a) Given that f is one-to-one on its domain, find a formula for the inverse
function f 1 x .
b) State the domain and range of f x and f 1 x .
SEQUENCES AND INDUCTION
2. Consider the sequence an given by a1 1, an1 1 2an , n 1.
a) Prove that 1 an 3 for all positive integers n.
b) Prove that an is increasing.
c) Explain how you know that the sequence converges, and find the limit of the
sequence.
3. Use induction to show that:
1 1 1 1 1
+ 2 + 3 + ⋯+ 𝑛 = 1 − 𝑛
2 2 2 2 2
for every integer 𝑛 ≥ 1.
LIMITS
4. Calculate the following limits if they exist. For any that don’t exist, explain why.
x 2 16
a) lim
x 4 tan 4 x
x 1
b) lim
x 1
x2 1
INVERSE FUNCTIONS
1. Consider the function given by f x ln x3 1 .
a) Given that f is one-to-one on its domain, find a formula for the inverse
function f 1 x .
b) State the domain and range of f x and f 1 x .
SEQUENCES AND INDUCTION
2. Consider the sequence an given by a1 1, an1 1 2an , n 1.
a) Prove that 1 an 3 for all positive integers n.
b) Prove that an is increasing.
c) Explain how you know that the sequence converges, and find the limit of the
sequence.
3. Use induction to show that:
1 1 1 1 1
+ 2 + 3 + ⋯+ 𝑛 = 1 − 𝑛
2 2 2 2 2
for every integer 𝑛 ≥ 1.
LIMITS
4. Calculate the following limits if they exist. For any that don’t exist, explain why.
x 2 16
a) lim
x 4 tan 4 x
x 1
b) lim
x 1
x2 1
