Q. 5 Explain why the limits聽limh鈫�0... [FREE SOLUTION]

Chapter 2: Q. 5 (page 183)

Explain why the limits $\lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h}$ and $\lim_{z 鈫� x} \frac{f (z) - f (x)}{z - x}$ are the same for any function $f$ . (Hint: Consider the substitution $z = x + h$ .)

Short Answer

Expert verified

$\lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h}$ and $\lim_{z 鈫� x} \frac{f (z) - f (x)}{z - x}$ are the same for any function $f$ .

Step by step solution

Step 1. Given information

We have to prove that $\lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h}$ and role="math" localid="1648378255156" $\lim_{z 鈫� x} \frac{f (z) - f (x)}{z - x}$ are the same for any function $f$ .

Step 2. Proof of the question

Substitute $z = x + h$

When $z 鈫� x$ , the value of $h 鈫� 0$

$\lim_{z 鈫� x} \frac{f (z) - f (x)}{z - x} = \lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{x + h - x} = \lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h}$

Hence it is proved.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Explain why the limits $\lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h}$ and $\lim_{z 鈫� x} \frac{f (z) - f (x)}{z - x}$ are the same for any function $f$ . (Hint: Consider the substitution $z = x + h$ .)

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof of the question

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Logic and Functions

Theoretical and Mathematical Physics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Explain why the limitslimh鈫�0fx+h-fxhandlimz鈫�xfz-fxz-xare the same for any function f. (Hint: Consider the substitution z=x+h.)

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof of the question

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Logic and Functions

Theoretical and Mathematical Physics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Explain why the limits $\lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h}$ and $\lim_{z 鈫� x} \frac{f (z) - f (x)}{z - x}$ are the same for any function $f$ . (Hint: Consider the substitution $z = x + h$ .)