What are some sample limit problems?
1 Answer
A few examples...
Explanation:
Limit problems can take many forms, so I will just give a few examples in increasing order of complexity...
Polynomials
Polynomials are always defined and continuous everywhere. Hence if
#lim_(x>a) f(x) = f(a)#
The behaviour as
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

#lim_(x>1) f(x)# 
#lim_(x>oo) f(x)# 
#lim_(x>+oo) f(x)#
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
Example:
Given:
#f(x) = (x^21)/(x^2+x2)#
what is
We find:
#f(x) = (x^21)/(x^2+x2) = (color(red)(cancel(color(black)((x1))))(x+1))/(color(red)(cancel(color(black)((x1))))(x+2)) = (x+1)/(x+2)#
with exclusion
So:
#lim_(x>1) f(x) = lim_(x>1) (x+1)/(x+2) = (color(blue)(1)+1)/(color(blue)(1)+2) = 2/3#
Radical functions
Limit problems involving radicals can often be addressed by multiplying by radical conjugates.
Example:
Given:
#f(x) = (sqrt(x+1)sqrt(x1))sqrt(x)#
What is
We find:
#lim_(x>oo) (sqrt(x+1)sqrt(x1))sqrt(x)#
#= lim_(x>oo) ((sqrt(x+1)sqrt(x1))(sqrt(x+1)+sqrt(x1))sqrt(x))/(sqrt(x+1)+sqrt(x1))#
#= lim_(x>oo) (((x+1)(x1))color(red)(cancel(color(black)(sqrt(x)))))/((sqrt(1+1/x)+sqrt(11/x))color(red)(cancel(color(black)(sqrt(x)))))#
#= lim_(x>oo) 2/(sqrt(1+1/x)+sqrt(11/x))#
#= 2/(sqrt(1+0)+sqrt(10))#
#= 2/2#
#= 1#