How do I tackle the true or false question: If lim x → 5 f(x) = 2 and lim x → 5 g(x) = 0, then lim x → 5 [f(x)/g(x)] does not exist.

Integrate

Integrate

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May 17, 2018
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At first I thought it I could create a rational function in which would create a removable discontinuity leaving only a numerator.

But I don't think it can be done.

I feel as if this question is pulling from limit laws.

Particularly, the quotient law.

If we let lim x → 5 f(x) be L and lim x → 5 f(x) = 0 be M that would mean the quotient law would be violated because M cannot equal zero.

But I don't feel like that's proof enough.

I feel as if I am missing something.

What am I missing about limits that should make this obvious for me?
 
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Try proving by contradiction: if [imath]h(x) = \frac{f(x)}{g(x)}[/imath], and [imath]h[/imath] has a finite limit [imath]A[/imath]: [imath]lim_{x\rightarrow 5} h(x) = A[/imath], then what can we say about the limit of [imath]g(x)h(x)[/imath] ?
 
blamocur said:
Try proving by contradiction: if [imath]h(x) = \frac{f(x)}{g(x)}[/imath], and [imath]h[/imath] has a finite limit [imath]A[/imath]: [imath]lim_{x\rightarrow 5} h(x) = A[/imath], then what can we say about the limit of [imath]g(x)h(x)[/imath] ?
Click to expand...
According to the product law of limits product law.png


Letting g(x) be f(x) and h(x) be g(x) that would mean that limx→5g(x)=A=M and limx→5f(x)=L therefore limx→5g(x)*limx→5f(x) = A*M
 
You can always use the squeeze theorem. You can find some neighborhood around x=5, where f(x) would be between 1.5 and 2.5.

1.5/g(x) < f(x)/g(x)< 2.5/g(x)

Now compute the lim as x goes to 5 on all three sides. The out limits will be undefined.
 
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