How do you evaluate the following limit: #lim_((x,y,z)->(-1,0,4)) ((x^3-ze^(2y))/(6x+2y-3z))#?

1 Answer
Kevin B.
Jun 22, 2017

Using direct substitution, you should get the answer #5/18#.

Explanation:

Given problem: #color(blue)(lim_((x,y,z)->(-1,0,4)) ((x^3-ze^(2y))/(6x+2y-3z))#

First thing we should always try is direct substitution. This means we just take the values that each variable is approaching and plug them into the expression.

#(((-1)^3-(4)e^(2(0)))/(6(-1)+2(0)-3(4))) =#

#(((-1)-(4))/((6(-1))-3(4)))) = #

#((-1-4)/(-6-12)) = color(blue)(5/18)#

This is the value that the function approaches as #(x,y,z)# approaches #(-1,0,4)#.

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