For what values of x, if any, does #f(x) = 1/((x-3)(x+4)(e^x-x)) # have vertical asymptotes?

Answer 1

The graph depicts the answer: x = 3 and #x=-4#.

#e^x-x# has the minimum 1, at x = 0.
So, #y to +-oo#, as #x to 3 and x to -4#.
And so, the vertical asymptotes are given by x = 3 and #x = - 4#.
The y-intercept of the graph is #-1/12# and it is the local minimum.

graph{y(x-3)(x+4)(e^x-x)-1=0 [-5, 5, -2.5, 2.5]}

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Answer 2

The function f(x) has vertical asymptotes at x = 3, x = -4, and at any value of x where e^x = x.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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