(a) Show that for 0 &lt

1. Homework 6 (1) In this exercise, we are going to prove lim

x→2x²= 4.

(a) Show that for 0 < |x − 2| < 1,

|x²− 4| < 5|x − 2|.

(b) Use (a) to prove the result.

(2) In this exercise, we are going to prove lim

x→3

1 x= 1

3. (a) Show that for 0 < |x − 3| < 1,

    1 x−1

3    

< 1 6|x − 3|.

(b) Use (a) to prove the result.

(3) Use definition to prove that lim

x→2

√ x

x²+ 5 = 2 3. (4) Suppose lim

x→af (x) = A and lim

x→ag(x) = B.

(a) Show that there exists d > 0 such that |g(x)| ≤ (|B| + 1) for 0 < |x − a| < d.

(b) Use (a) to show that

|f (x)g(x) − AB| ≤ (|B| + 1)|f (x) − A| + |A||g(x) − B|

for 0 < |x − a| < d.

(c) Use (b) to show that lim

x→af (x)g(x) = AB.

(d) Suppose B 6= 0. Show that there exists d⁰> 0 such that

|g(x)| > |B|/2, 0 < |x − a| < d⁰. (e) Use (d) to show that

1 g(x)− 1

B    

< 2

B²|g(x) − B|

for 0 < |x − a| < d⁰. (f) Use (e) to show that lim

x→a

1 g(x)= 1

B. (g) Use (c) and (f) to show that lim

x→a

f (x) g(x) = A

B provided that B 6= 0.

(h) Use induction and (c) to prove that lim

x→af (x)ⁿ= Aⁿ for any n ∈ N.

(5) Let n ≥ 2 be a natural number and a > 0. Denote M = aⁿ⁻¹ⁿ

2(1 − qⁿ

1 2)

.

(a) Show that if |x − a| < a/2, we have

xⁿ⁻¹ⁿ + aⁿ¹xⁿ⁻²ⁿ + · · · + aⁿ⁻²ⁿ xⁿ¹ + aⁿ⁻¹ⁿ > M.

(b) Show that if |x − a| < a/2, we have

|√ⁿ x −√

a| < M⁻¹|x − a|.

(c) Show that lim

x→a

√n

x = √ⁿ a.

(6) (Sandwich Principle) Suppose that f, g, h are functions defined on 0 < |x − a| < d for some d > 0 and

h(x) ≤ f (x) ≤ g(x), 0 < |x − a| < d.

If lim

x→ah(x) = lim

x→ag(x) = L, show that lim

x→af (x) = L. Use this to evaluate lim

x→0

p3

x + x²sin1 x.

1

2

(7) Let f (x) be a real-valued function on R defined by f (x) =

(_{sin x}2

x² if x 6= 0, 1 otherwise.

Is f (x) continuous at 0?

(8) Let f (x) and g(x) be real valued functions on R defined below.

f (x) =

(0 if x is rational,

1 if x is irrational., g(x) =

(0 if x is rational, x if x is irrational.

Evaluate lim

x→0f (x) and lim

x→0g(x) if they exist.

(9) Find the value a such that lim

x→0

√3

ax + 8 − 2

x = 5

12. (10) Let p(x) = x⁶+ 2x⁵− x + 1. Evaluate lim

x→∞(p⁶

p(x) − x).

(11) Evaluate the following limits.

(a) lim

x→3

√x + 6 − x x³− 3x² . (b) lim

x→0

(1 + x)(1 + 2x)(1 + 3x) − 1

x .

(c) lim

x→1

x¹⁰− 1 x⁹⁹− 1. (d) lim

x→2

(x¹⁰− 2¹⁰) − 10 · 2⁹(x − 2)

(x − 2)² .

(e) lim

x→1

 m

1 − x^m− n 1 − xⁿ



where m, n are distinct natural numbers.

(f) lim

x→−2

√3

x − 6 + 2 x³+ 8 . (g) lim

x→1

m√ x − 1

√n

x − 1, where m, n are natural numbers.

(h) lim

x→0

sin 20x

x .

(i) lim

x→0

sin 5x sin 7x. (j) lim

x→0

1 − cos x x² . (k) lim

x→0

sin 6x − sin 5x sin x .