If # f(x) = { (x^2, x !=2), (2, x=2) :} # then evaluate # lim_(x rarr 2) f(x)#?

Question

If #  f(x) = { (x^2, x !=2), (2, x=2) :} # then evaluate # lim_(x rarr 2) f(x)#?

Charles Anctil · Accepted Answer

# lim_(x rarr 2) f(x) = 4#

In order to evaluate a limit we are not interested in the value of the function at the limit, just the behaviour of the function around the limit: We have: # f(x) = { (x^2, x !=2), (2, x=2) :} # If we graphed the function would be the parabola #y=x^2# with a hole at #x=2#. If we just examine the behaviour close to #x=2#, e.g #x=2+-0.001# we get: # f(2-0.001) = f(1.999) = 3.996001 # # f(2+0.001) = f(2.001) = 4.004001 # # f(2) = 2 # Which would certainly "suggest" that #lim_(x rarr 2) f(x)# is #4# rather than #2#. We can show thi is the case analytically as follows by studying the limit either side of #x=2#: The Left Handed limit: # lim_(x rarr 2^-) f(x) = lim_(x rarr 2^-) x^2# # " " = 2^2# # " " = 4# And The Right Handed limit: # lim_(x rarr 2^+) f(x) = lim_(x rarr 2^+) x^2# # " " = 2^2# # " " = 4# And as both limits are identical we have: # lim_(x rarr 2^) f(x) = 4# In both cases we chose the "#x^2#" variant of the function as #x rarr 2# but #x != 2#

Cameron Alexander · Answer

Please see the discussion below.

One way to think about the limit of a function as #x# approaches #2# is that it is the value you would expect if you looked at the graph but covered up the graph where #x = 2#. The limit wants to know "for values of #x# near #2#, is the value of #f(x)# (think of it as #y#) close to just one number. And if so, what number is that? The limit, #lim_(xrarr2)f(x)# doesn't care what happens when #x = 2#, just what happens when #x# is close to #2# The question of what happens whar #x# equals two is the question of the value of #f(2)#. In your example, #f(2) = 4# but for #x# near #2# (and not equal to #2#) #f(x) = x^2# is close to #4#. So that is the limit. Finally, functions like the one in this question (kind of strange, with weird points) are very important to learning the difference between: "Find #f(a)#" and "Find #lim_(xrarra)f(x)#"

Emma Aldridge · Answer

undefined The limit of f(x) as x approaches 2 is equal to 4.