Lim x>0 (sqr5+x-sqr5)/x
First step in evaluating limits:
simplify the algebra and see what you get.
If you get:
(a number), then it is also the limit.
(∞/(a number)), the limit is &infin.
(0/(a number)), the limit is 0.
(∞/∞), or (0/0), then you need other techniques/tools.
Try the first step shown above and tell us what you get.
some parentheses would also help.
Since (√5 + x - √5)/x = 1, that's probably not what you had in mind.
(√(5+x)-√5)/x -> 0/0
so, multiply top and bottom by (√(5+x)+√5) and you end up with
((5+x)-5)/(x(√(5+x)+√5))
see where that takes you...
To simplify the given expression, let's start by simplifying the numerator:
√5 + x - √5
Notice that the square root terms cancel each other out:
(√5 - √5) + x
Since (√5 - √5) equals zero, we have:
0 + x
Which simplifies to:
x
So, the simplified expression is just x.
Now, let's look at the limit as x approaches 0:
lim x→0 x
Since x is a linear term, the limit of x as x approaches 0 is simply 0.
Therefore, the value of the given expression, lim x→0 (√5 + x - √5)/x, is 0.