How do you find the limit of #\lim _ { x \rightarrow - \infty } ( \frac { x } { e ^ { x } - x } )#?

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Answer 1 · 2017-06-26T15:34:46

#lim_(x->-oo) x/(e^x-x) = -1#

Write the limit as:

#lim_(x->-oo) x/(e^x-x) = lim_(x->-oo) 1/((e^x-x)/x) = lim_(x->-oo) 1/(e^x/x-1)#

As:

#lim_(x->-oo) e^x/x = lim_(x->-oo) 1/x xx lim_(x->-oo) e^x = 0#

then:

#lim_(x->-oo) x/(e^x-x) = 1/(0-1) = -1#

graph{x/(e^x-x) [-10, 10, -5, 5]}