Limit Definition of Derivative
Key Questions

Limit Definition of
#f'(a)# #f'(a)=lim_{h to 0}{f(a+h)f(a)}/h# or
#f'(a)=lim_{x to a}{f(x)f(a)}/{xa}# 
#f(x)=c# is a constant function, so its value stays the same regardless of the xvalue. In particular,#f(x+h)=c# .By the definition of the derivative,
#f'(x)=lim_{h to 0}{f(x+h)f(x)}/h# #=lim_{h to 0}{cc}/{h}# #=lim_{h to 0}0# #=0# 
Remember that the limit definition of the derivative goes like this:
#f'(x)=lim_{h rightarrow0}{f(x+h)f(x)}/{h}# .
So, for the posted function, we have
#f'(x)=lim_{hrightarrow0}{m(x+h)+b[mx+b]}/{h}#
By multiplying out the numerator,
#=lim_{hrightarrow0}{mx+mh+bmxb}/{h}#
By cancelling out#mx# 's and#b# 's,
#=lim_{hrightarrow0}{mh}/{h}#
By cancellng out#h# 's,
#=lim_{hrightarrow0}m=m#
Hence,#f'(x)=m# .The answer above makes sense since the derivative tells us about the slope of the tangent line to the graph of
#f# , and the slope of the linear function (its graph is a line) is#m# . 
Yes, there is a difference since the first limit is defined at
#x=0# , but the second one is not.
I hope that this was helpful.
Questions
Derivatives

Tangent Line to a Curve

Normal Line to a Tangent

Slope of a Curve at a Point

Average Velocity

Instantaneous Velocity

Limit Definition of Derivative

First Principles Example 1: x²

First Principles Example 2: x³

First Principles Example 3: square root of x

Standard Notation and Terminology

Differentiable vs. Nondifferentiable Functions

Rate of Change of a Function

Average Rate of Change Over an Interval

Instantaneous Rate of Change at a Point