Math Test: Limits, Derivatives & Combination/Permutation

[mohlam12]

Hey!
So monday I have my math test, it's going to be about limits, derivatives and combination & permutation ...
I have some question;
if you get an indeterminate form of +$$\infty - \infty$$, is there any rule to ease everything for you. like how you can use the Hopital ruel for 0/0 ? anw what do you do in general when you get this kind of indeterminate form?
if they ask you to see if the function is derivable in $$\x_{0}$$, when do you study the right and left derivation ?
that s it for now!
hope to hear from you soon. Thanks
Mohammed

 

[VietDao29]

Hey!
So monday I have my math test, it's going to be about limits, derivatives and combination & permutation ...
I have some question;
if you get an indeterminate form of +$$\infty - \infty$$, is there any rule to ease everything for you. like how you can use the Hopital ruel for 0/0 ? anw what do you do in general when you get this kind of indeterminate form?

Uhmm, can you be little bit clearer? Like giving us an example, or something along those lines...
If you were ask to find the limit of some radicals, then it's common to multiply and divide the whole thing by its conjugate expression.
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Example:
Find
$$\lim_{x \rightarrow +\infty} \sqrt{x ^ 2 + x} - x$$
Now, it's the Indeterminate form $$+ \infty - \infty$$, right?
Multiply the whole expression with $$\frac{\sqrt{x ^ 2 + x} + x}{\sqrt{x ^ 2 + x} + x}$$ to obtain:
$$\lim_{x \rightarrow +\infty} \sqrt{x ^ 2 + x} - x = \lim_{x \rightarrow +\infty} \frac{(\sqrt{x ^ 2 + x} - x) (\sqrt{x ^ 2 + x} + x)}{\sqrt{x ^ 2 + x} + x}$$
$$= \lim_{x \rightarrow +\infty} \frac{x ^ 2 + x - x ^ 2}{\sqrt{x ^ 2 + x} + x} = \lim_{x \rightarrow +\infty} \frac{x}{\sqrt{x ^ 2 + x} + x}$$
Now divide both numerator and denominator by x, note that as x tends to positive infinity, then x > 0, so $$x = \sqrt{x ^ 2}$$
$$\lim_{x \rightarrow +\infty} \frac{x}{\sqrt{x ^ 2 + x} + x} = \lim_{x \rightarrow +\infty} \frac{1}{\sqrt{1 + \frac{1}{x}} + 1} = \frac{1}{2}$$.
Can you get this? :)

if they ask you to see if the function is derivable in $$\x_{0}$$, when do you study the right and left derivation ?
that s it for now!
hope to hear from you soon. Thanks
Mohammed

Do you mean differentiable? Yes, kind of, if f(x₀) is differentiable at x₀, then we must have:
$$f'(x_0 ^ +) = f'(x_0 ^ -)$$.
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By the way, we don't need a slash before an x in your LaTeX code, we just need to type x_0, backslashes are used to make functions displayed in normal font (i.e not in italics) so that they'll stand out from the rest (which is in italic font).
Example: \sin x
$$\sin x$$

 

[Architeuthis Dux]

Hi Mohammed,

It sounds like you have a lot of great questions about your upcoming math test. For the indeterminate form of +\infty - \infty, there is a rule called the "limit of the difference equals the difference of the limits" which can be used to simplify the expression. However, if this rule doesn't work for your specific problem, you can also try using L'Hopital's rule, which you mentioned. This rule states that if you have a limit of the form 0/0 or \infty/\infty, you can take the derivative of the numerator and denominator separately and then evaluate the limit again. As for when to use right and left derivatives, it depends on the function and the specific point you are studying. In general, you should use the right derivative when approaching the point from the right side and the left derivative when approaching from the left side. I hope this helps and good luck on your test!