# The lim x → 0 9 x − 5 x x . Also confirm the result graphically.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2.2, Problem 24E
To determine

Expert Solution

## Answer to Problem 24E

limx09x5xx=0.588 .

### Explanation of Solution

Given information:

Given limit as limx09x5xx

Consider the limits as, limx09x5xx

Since, ax=1+xlna+(xlna)22!+(xlna)33!+...

Hence,

lim x0 9 x 5 x x = lim x0 1+xln9+ ( xln9 ) 2 2! + ( xln9 ) 3 3! +...( 1+xln5+ ( xln5 ) 2 2! + ( xln5 ) 3 3! +... ) x

= lim x0 1+xln9+ ( xln9 ) 2 2! + ( xln9 ) 3 3! +...1xln5 ( xln5 ) 2 2! ( xln5 ) 3 3! ... x

= lim x0 ( xln9xln5 )+ 1 2 ( ( xln9 ) 2 ( xln5 ) 2 )+ 1 6 ( ( xln9 ) 3 ( xln5 ) 3 )+... x

= lim x0 x( ( ln9ln5 )+ x 2 ( ( ln9 ) 2 ( ln5 ) 2 )+ x 2 6 ( ( ln9 ) 3 ( ln5 ) 3 )+... ) x

= lim x0 ( ( ln9ln5 )+ x 2 ( ( ln9 ) 2 ( ln5 ) 2 )+ x 2 6 ( ( ln9 ) 3 ( ln5 ) 3 )+... )

=ln 9 5

=0.588

Graph of the function is,

Therefore, limx09x5xx=0.588

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