LIMITS
Fig. 6-3
6.40
Evaluate
As x—»+00, both VoT + x and x approach +*. It is not obvious how their difference behaves.
However, the limit equals
6.42
Let f(x) = a n x" + a n _ l x" ' + ••• + a,x + a a , with a,, >0. Prove that lim f(x) = +».
6.44
If
-2- with «,,>0 and b k >0, prove that lim f(x) = +» if n >k.
Factoring out x" from the numerator and then dividing numerator and denominator by x
k , f(x) becomes
As *-»+oo, all the quotients a n _ j lx' and b k _ i lx
l approach 0, and, therefore, the quantity inside the
parentheses approaches a n /b k >Q. Since x"~
k approaches +00, lim f(x)=+
x -
Now we divide numerator and denominator by x (noting that
6.41
Evaluate
Rationalize the numerator
We obtain
Answer
sum inside the parentheses approaches
with a
show that
6.43
Since each «„_,/*' and b n _ i /x' approaches 0,
Dividing numerator and denominator by x",
41