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: \(\displaystyle \lim_{{x}\to{0^-}}\frac{1}{x}\)
What do you get when you type in \(\displaystyle \frac{1}{-.5}\) on your calculator? What about \(\displaystyle \frac{1}{-.1}\)? \(\displaystyle \frac{1}{-.01}\)? 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.
The left hand limit at infinity refers to the behavior of a function as the input values approach negative infinity, while the right hand limit at infinity refers to the behavior as the input values approach positive infinity.
To determine the limit at infinity of a function, you can evaluate the limit as the input values approach infinity. This can be done by finding the horizontal asymptote of the function or by using L'Hopital's rule.
Yes, a function can have different left and right hand limits at infinity if the behavior of the function is different as the input values approach negative infinity and positive infinity.
If a function has a finite limit at infinity, it means that as the input values approach infinity, the output values of the function approach a specific finite number. This can be represented as lim f(x) = L as x approaches infinity.
The behavior of a function at infinity can affect its graph in different ways. For example, if a function has a finite limit at infinity, its graph will approach a horizontal asymptote. If a function has an infinite limit at infinity, its graph will approach vertical asymptotes. If a function has no limit at infinity, its graph will have a type of discontinuity at infinity.