Limit of function using epsilon delta proof

Question

lim 
x → 2 
(x^3 − 2x + 1) = 5 

illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1. (Round your answers to four decimal places.) 



ε = 0.2 δ = ? 
ε = 0.1 δ = ?

Answer 1

(a)

Step 1:

The limit of the function is .

definition of limit : ,

if for every number of , there exists a such that whenever .

From the definition, if then .

Step 2:

For :

Consider

.

Graph :

Graph the function :

Observe the graph:

lies in the interval .

Step 3:

Consider .

.

lies in the interval .

Compare with .

Here value should be a smaller positive number.

Therefore .

Solution:

for .

Answer 2

(b)

Step 1:

From definition of limit: if then .

For :

Consider

.

Graph :

Graph the function :

Observe the graph:

lies in the interval .

Step 2:

Consider .

.

lies in the interval .

.

Compare with .

Here value should be a smaller positive number.

Therefore, .

Solution:

for .