The Student Room Group

Limit Question

Hey guys, I am so confused by this question so some help would be appreciated :smile:

For what values of a and b does limxπf(x)\lim_{x \rightarrow \pi}f(x) exist?

f(x)=1x1+bf(x)=\frac{1}{x-1}+b for x<πx<\pi, where defined
f(x)=af(x)=a for x=π x=\pi
f(x)=tan(x2π)f(x)=tan(\frac{x^2}{\pi}) for x>πx>\pi, where defined.

Thanks!
This is a really nice question.
Let's talk about aa first. Imagine that you have the function
[br]f(x)={1if x0[br]0otherwise .[br] [br]f(x) = \begin{cases} 1 &\text{if } x \neq 0 \\ [br] 0 & \text{otherwise }. \end{cases}[br].
Does the limit as x0 x \to 0 exist? If so, why? If not, why not?
Reply 2
Original post by IrrationalNumber
This is a really nice question.
Let's talk about aa first. Imagine that you have the function
[br]f(x)={1if x0[br]0otherwise .[br] [br]f(x) = \begin{cases} 1 &\text{if } x \neq 0 \\ [br] 0 & \text{otherwise }. \end{cases}[br].
Does the limit as x0 x \to 0 exist? If so, why? If not, why not?


surely its just 0?
Reply 3
Original post by IceKidd
surely its just 0?
What is your definition of limit?
Original post by IceKidd
surely its just 0?


Generally, lim f(x) =f (lim x) only when the function is continuous.

In the example he gave, there's no delta value for quite a lot of epsilons for a corresponding limit of zero.
Reply 5
Sorry guys, I am totally clueless about this. Any more helpful hints to guide me along the way?
Reply 6
One thing our lecturer told us that's useful to remember, when taking limits we do not care what the value is at the point we are approaching, just how it gets there. For example if you had a line that was just 1 everywhere except at 0, then the limit as x->0 is 1 and it does not matter what the value at 0 is.
Original post by james22
One thing our lecturer told us that's useful to remember, when taking limits we do not care what the value is at the point we are approaching, just how it gets there. For example if you had a line that was just 1 everywhere except at 0, then the limit as x->0 is 1 and it does not matter what the value at 0 is.


Bingo! So, OP, with this in mind, what values can a a take?

(In fact, this is the example in post #2)

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