[Math] Finding Limit Function that satisfies the conditions.

calculus

I am having some trouble figuring out a few math problems from my Calc 1 class. I am not sure where to start, as all the limits are different.
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  1. find a function that satisfies the given conditions and then sketch it.

  2. sketch a graph of the function y=f(x) that satisfies the given conditions. Just label the coordinate axes and sketch the appropriate graph.

For 60, 62, and 64 they are kinda the same thing.
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60: would it be a function where if you let X=-1, and the denominator =0, is that what we are looking for? like $ \frac{2}{x+1} \ . $

  1. I would say yes, even though g(x) and f(x) are not discontinuous on their own, that changes when you put them in a function together such as $ f(x)/g(x).$

  2. not to sure

Thank you for all your help.

Best Answer

Just try drawing what the graph could potentially be on paper, and find an equation for it. Here's a freebie for the first one (number 74): $$g(x)=\dfrac{1}{x-3}$$ Note: I am not taking calculus yet, but I do know about limits and such... I don't know if there is a faster way because my way seems inefficient and not algebraic-like.

EDIT: For the second one, maybe this would work: $$f(x)=\begin{cases} -\frac{1}{x^2+\frac{1}{2}}, \ \ \ \ x < 0 \\ 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=0 \\ \frac{1}{x^2+\frac{1}{2}}, \ \ \ \ \ \ \ \ x > 0 \\ \end{cases}$$ YET ANOTHER EDIT: Your answer for the final one (nonremovable discontinuity) is correct.

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