(1/t√1+t - 1 /t) lim

t➡️0

So far I factored out 1/t√1+t - 1/t x √1+t/√1+t

I am having trouble going further

Hard to make out what you mean without brackets for the denominators

the way you typed it....

1/t√1 is just 1/t, remember terms are separated by addition or subtraction

so you simply have
1/t + t - 1/t
= t

I am sure that is not you meant.

No its written as 1 in the numerator then in the denominator it is written as t√1+t this is then subtracted by 1/t lim is t----> 0

How would you factor this?

Still not clear enough

if it would be t√1 + t
why would they put in that √1 ?
it would simply become t(1) + t
or 2t

is it
t√(1+t) ?

there is a monumental difference

should I guess at

Limit (1/( t√(1+t) - 1/t ) , as x ---> 0 ???

yes that is what it is!! sorry for the confusion

OK THEN

let's add up 1/( t√(1+t) - 1/t
= (1 - √(1+t)/(t√(1+t)
= (1 - √(1+t)/(t√(1+t) * (1 + √(1+t) / (1 + √(1+t)
= (1 - 1 - t)/(t√(1+t)((1 + √(1+t) )
= -t/(t√(1+t)((1 + √(1+t) )
= -1/(√(1+t)((1 + √(1+t) )
so lim -1/(√(1+t)((1 + √(1+t) ) as t ---> 0
= -1/(√1(1 + √1)
= -1/2

To further simplify the expression, we can rationalize the numerator. Here's how:

Starting with your expression:

(1/t√(1+t) - 1/t)

First, let's rewrite the expression by finding a common denominator for the two terms in the numerator:

= [1/ (t√(1+t)) - (1/t)] x √(1+t)/√(1+t)

Now, we can combine the terms in the numerator:

= [1 - (t√(1+t))/(t√(1+t))] x √(1+t)/√(1+t)

Simplifying further:

= [(1 - t√(1+t))/(t√(1+t))] x √(1+t)/√(1+t)

Now, we can cancel out the common factors in the numerator and denominator:

= (1 - t√(1+t))/t

Finally, we can simplify the expression by taking the limit as t approaches 0.

As t approaches 0, the term t√(1+t) goes to 0, and the expression becomes:

= (1 - 0)/0

However, we cannot divide by 0. Therefore, the limit of the given expression does not exist.