What are some sample limit problems?

Limit problems can take many forms, so I will just give a few examples in increasing order of complexity...

#color(white)()#
Polynomials

Polynomials are always defined and continuous everywhere. Hence if #f(x)# is a polynomial and #a in RR#, then:

#lim_(x->a) f(x) = f(a)#

The behaviour as #x->+oo# or #x->-oo# is determined solely by the leading term (i.e. term of highest degree), which will dominate the other terms when #x# is sufficiently large.

Hence we get rules:

• If #f(x)# is of odd degree with positive leading coefficient then:#color(white)(0/0)#
  #lim_(x->-oo) f(x) = -oo# and #lim_(x->+oo) f(x) = +oo#

• If #f(x)# is of odd degree with negative leading coefficient then:#color(white)(0/0)#
  #lim_(x->-oo) f(x) = +oo# and #lim_(x->-oo) f(x) = -oo#

• If #f(x)# is of even degree with positive leading coefficient then:#color(white)(0/0)#
  #lim_(x->-oo) f(x) = +oo# and #lim_(x->+oo) f(x) = +oo#

• If #f(x)# is of even degree with negative leading coefficient then:#color(white)(0/0)#
  #lim_(x->-oo) f(x) = -oo# and #lim_(x->+oo) f(x) = -oo#

Example:

Given #f(x) = -2x^3+5x+7# evaluate:

• #lim_(x->1) f(x)#

• #lim_(x->-oo) f(x)#

• #lim_(x->+oo) f(x)#

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Rational functions

Like polynomials these are continuous everywhere, except when the denominator is zero. Typical problems might involve evaluating a limit where both numerator and denominator tend to #0#. These usually involve simplifying the rational expression by identifying common factors.

Example:

Given:

#f(x) = (x^2-1)/(x^2+x-2)#

what is #lim_(x->1) f(x)# ?

We find:

#f(x) = (x^2-1)/(x^2+x-2) = (color(red)(cancel(color(black)((x-1))))(x+1))/(color(red)(cancel(color(black)((x-1))))(x+2)) = (x+1)/(x+2)#

with exclusion #x != 1#

So:

#lim_(x->1) f(x) = lim_(x->1) (x+1)/(x+2) = (color(blue)(1)+1)/(color(blue)(1)+2) = 2/3#

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Radical functions

Limit problems involving radicals can often be addressed by multiplying by radical conjugates.

Example:

Given:

#f(x) = (sqrt(x+1)-sqrt(x-1))sqrt(x)#

What is #lim_(x->oo) f(x)# ?

We find:

#lim_(x->oo) (sqrt(x+1)-sqrt(x-1))sqrt(x)#

#= lim_(x->oo) ((sqrt(x+1)-sqrt(x-1))(sqrt(x+1)+sqrt(x-1))sqrt(x))/(sqrt(x+1)+sqrt(x-1))#

#= lim_(x->oo) (((x+1)-(x-1))color(red)(cancel(color(black)(sqrt(x)))))/((sqrt(1+1/x)+sqrt(1-1/x))color(red)(cancel(color(black)(sqrt(x)))))#

#= lim_(x->oo) 2/(sqrt(1+1/x)+sqrt(1-1/x))#

#= 2/(sqrt(1+0)+sqrt(1-0))#

#= 2/2#

#= 1#