Limit of Greatest Integer Function...7

[nycmathguy]

Extra practice not from the textbook.

Find the limit of

f(x) = [[ x^2 ]]^(2x - 1) + [[ 1/x ]]^([[ 1/x^2 ]]

as x tends to zero.

How is this done? Can you show the graph of f(x)?

 

[MathLover1]

lim( [[ x^2 ]]^(2x - 1) + [[ 1/x ]]^([[ 1/x^2 ]]) as x->0

lim( [[ x^2 ]]^(2x - 1)) + lim([[ 1/x ]]^([[ 1/x^2 ]])

substitute x
lim( [[ 0^2 ]]^(2*0 - 1)) + lim([[ 1/0 ]]^([[ 1/0^2 ]])

lim((0^2)^( - 1)) + lim([[ 1/0 ]]^( 1/0^2)=1/0^2 +(1/0)^(1/0)=∞

so,

 

[nycmathguy]

lim( [[ x^2 ]]^(2x - 1) + [[ 1/x ]]^([[ 1/x^2 ]]) as x->0

lim( [[ x^2 ]]^(2x - 1)) + lim([[ 1/x ]]^([[ 1/x^2 ]])

substitute x
lim( [[ 0^2 ]]^(2*0 - 1)) + lim([[ 1/0 ]]^([[ 1/0^2 ]])

lim((0^2)^( - 1)) + lim([[ 1/0 ]]^( 1/0^2)=1/0^2 +(1/0)^(1/0)=∞

so,
View attachment 2662

Thank you. This one threw me into a loop-o-plane.