The limit of the function $\frac{1}{x}$ as $x$ approaches 0 does not exist due to differing behaviors from the left-hand limit (LHL) and right-hand limit (RHL). The left-hand limit, $\lim_{{x}\to{0^-}}\frac{1}{x}$, approaches negative infinity, while the right-hand limit, $\lim_{{x}\to{0^+}}\frac{1}{x}$, approaches positive infinity. Therefore, since the LHL and RHL do not converge to the same value, the overall limit does not exist.
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pcforgeek said:Show that $\lim_{{x}\to{0}}$ $\frac{1}{x}$ does not exist?
Please tell me how to make LHL and RHL for this? Explain me all steps used?
Jameson said:Hi pcforgeek, (Wave)
Welcome to MHB!
We don't really give out answers or do problems fully for others, rather help you solve it yourself. So let's do that. :)
Ok, starting from the left: $$\lim_{{x}\to{0^-}}\frac{1}{x}$$
What do you get when you type in $$\frac{1}{-.5}$$ on your calculator? What about $$\frac{1}{-.1}$$? $$\frac{1}{-.01}$$? Notice that these numbers are walking closer and close to 0 from the negative side of it. Any idea where this trend is going?
pcforgeek said:I already know that as the denominator becomes smaller and smaller and closer to zero the value becomes infinity. Can you tell me how to get right hand limit and left hand limit for this problem.
Jameson said:I was showing you how to calculate the left hand limit actually, which does not approach infinity. Did you try what I asked? Try again and tell me what the trend is for values approaching 0 from the left. Plug in those values I suggested and you should see a pattern.
pcforgeek said:I finally understand now. LHL is negative infinity and RHL is positive infinity.