Chapter 1.2, Problem 48E

### Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361

Chapter
Section

### Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361
Textbook Problem

# Using the ε − δ Definition of Limit In Exercises 45-56, find the limit L. Then use the ε − δ definition to prove that the limit is L. lim x → − 3 ( 3 4 x + 1 )

To determine

To calculate: The value of limit L of a function limx334x+1 and proving the result by εδ definition.

Explanation

Given:

limxâ†’3â€‰â€‰34x+1

limxâ†’3â€‰34x+1=94+1=â€‰134

Calculation:

Îµâˆ’Î´ definition states that if âˆ€â€‰Îµ>0,â€‰â€‰thenâ€‰thereâ€‰existsâ€‰aâ€‰postiveâ€‰ integer Î´>0 such that |f(x)âˆ’L|<Îµâ€‰,â€‰â€‰whenâ€‰everâ€‰â€‰â€‰0<â€‰|xâˆ’c|<Î´ â€”â€”â€“(1)

Substituting f(x)=34x+1,â€‰â€‰L=134,â€‰â€‰andâ€‰â€‰c=3â€‰â€‰inâ€‰statementâ€‰(1),â€‰â€‰weâ€‰getâ€‰

|34x+1âˆ’134|<Îµâ€‰â€‰â€‰whenâ€‰everâ€‰â€‰â€‰0<â€‰|xâˆ’3|<Î´

â‡’|3xâˆ’94|<

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