Question

Limit-Can limit be squared on both sides?

Answer 1

hell yes ... but you can't take back square root

Answer 2

@experimentX
for example,
if lim(x-->0)sqrt(ax+b)=x+2
Can it be written as lim(x-->0)(ax+b)=(x+2)^2

Answer 3

hmm ... no!! you never get this type of limit.
lim(x-->0)sqrt(ax+b)=x+2
^^ there cannot be variable 'x' on this side.

Answer 4

\[\lim \limits_{x\to 0} (1+x)^ {\frac{1}{x}} = k\]

Answer 5

if u try to take x power both sides... im not sure.. it just doesnt look right

Answer 6

lol ... you can't take variable as ^ on both sides.

Answer 7

hahah yes :)

\[\large \lim \limits_{x\to 0} (1+x)^ {\frac{1}{n}} = k\]

Answer 8

taking nth power both sides is legal ?

Answer 9

this like proving 1=2 via derivative x^2

Answer 10

but not sure about 1/n th

Answer 11

Perhaps you can try for something more general. Suppose \(g\) is continuous and we have
\(\lim_{x \to n} f(x) = L\)
Does it hold that
\(\lim_{x \to n} g(f(x)) = g(L)\)

Answer 12

taking n-th power you get,
yes yes of course you can do it ...
\[ \lim_{x \to n} g(f(x)) = g \left( \lim_{x\to 0 }f(x) \right) = f(L)\]

Answer 13

sorry ... g(L)


but not sure about taking power 1/n th

Answer 14

@Alchemista yes it holds,we can multiply the same value on both sides.
But,it is confusing about the square?

Answer 15

Why? Let g(x) = x^2

Answer 16

define g(x) = x^2 in above, you get the result immediately.

Answer 17

You still need to prove the so called "composition law" of limits via first principles if you want a rigorous proof.

You will need to appeal to definition of limit, the epsilon-delta definition.

Answer 18

that limit works like a chain rule ... chain rule of logic.

Answer 19

It is indeed similar logic.