Formal Definition of a Limit at a Point
Key Questions

Answer:
See below
Explanation:
The definition of limit of a sequence is:
Given
#{a_n}# a sequence of real numbers, we say that#{a_n}# has limit#l# if and only if#AA epsilon>0, exists n_0 in NN // AAn>n_0 rArr abs(a_nl))< epsilon# 
Before writing a proof, I would do some scratch work in order to find the expression for
#delta# in terms of#epsilon# .According to the epsilon delta definition, we want to say:
For all
#epsilon > 0# , there exists#delta > 0# such that
#0<x1< delta Rightarrow (x+2)3 < epsilon# .Start with the conclusion.
#(x+2)3 < epsilon Leftrightarrow x1 < epsilon# So, it seems that we can set
#delta =epsilon# .(Note: The above observation is just for finding the expression for
#delta# , so you do not have to include it as a part of the proof.)Here is the actual proof:
Proof
For all
#epsilon > 0# , there exists#delta=epsilon > 0# such that
#0<x1 < delta Rightarrow x1< epsilon Rightarrow (x+2)3 < epsilon# 
Precise Definitions
Finite Limit
#lim_{x to a}f(x)=L# if
for all#epsilon>0# , there exists#delta>0# such that
#0<xa< delta Rightarrow f(x)L < epsilon# Infinite Limits
#lim_{x to a}f(x)=+infty# if
for all#M>0# , there exists#delta>0# such that
#0<xa< delta Rightarrow f(x)>M# #lim_{x to a}f(x)=infty# if
for all#N<0# , there exists#delta>0# such that
#0<xa< delta Rightarrow f(x) < N#
Questions
Limits

Introduction to Limits

Determining One Sided Limits

Determining When a Limit does not Exist

Determining Limits Algebraically

Infinite Limits and Vertical Asymptotes

Limits at Infinity and Horizontal Asymptotes

Definition of Continuity at a Point

Classifying Topics of Discontinuity (removable vs. nonremovable)

Determining Limits Graphically

Formal Definition of a Limit at a Point

Continuous Functions

Intemediate Value Theorem

Limits for The Squeeze Theorem