lim t -> 0 ((√(2+t) - √2) / t)

Question

how is the answer √2 / 4? I don't know what to do with the denominator (I already multiplied by the conjugate, which I put as √2+t + sqrt2) I still got different answers.

Thanks!

Answer 1

You are correct to use the conjugate √(2 + t) + √(2). Here's the derivation:

√(2+t) - √(2) √(2+t) + √(2) 2+t - 2 t

————————————— X ————————————— = ——————————————— = ———————————————

t √(2+t) + √(2) t√(2+t) + t√(2) t√(2+t) + t√(2)

Notice, in the final expression, every term in the numerator and denominator has a factor t, so we can divide out all t's. After doing so, plugging in 0 for t no longer makes the expression undefined, so we can solve the limit as follows:

1 1 1 1 √(2) √(2)

————————————— = ————————————— = ——————————— = ————— = ———— = ————

√(2+t) + √(2) √(2+0) + √(2) √(2) + √(2) 2√(2) 2(2) 4

This is how you arrive at the answer √(2) / 4. I don't know where you are in calculus/precalculus, but as a look towards the future, this is actually an example of a calculus concept called the derivative. The derivative of a function f(x) is defined as:

lim f(x+h) - f(x) lim f(2+t) - f(2)

x→0 ————————————— , which looks similar to t→0 ————————————— doesn't it?

h t

The derivative tells you the slope of a graph at any point along the curve, which, as you'll learn, can be useful to know. There are also shortcut rules to figuring out the derivative of a function. In the case of this problem, f(x) = √(x) or in other terms f(x) = x^1/2, and the general derivative is f '(x) = (1/2)x^-1/2. If you input 2 for x, like for this problem, you'll find that f '(2) evaluates to √(2) / 4, just like our solution.

Hope that answers your question and makes sense :)