Having a hard time solving for this limit.

Question

Answer 1

When limits have radicals like this, it's never a bad idea to rationalize or to multiply by a conjugate. We'll rationalize!

The second term in the limit (the 1 / t) should be multiplied by sqrt(1 + t) on top and bottom (i.e., by a factor of one). When you do that and simplify, your limit should read:

lim (t -> 0) [ (1 - sqrt(1 + t)) / (t*sqrt(1 + t)) ].

From here, apply L'Hopital's rule, so be diligent and careful with your derivatives (with chain rule!) in the numerator and denominator moving forward from here. If you do it correctly, the problem becomes much more manageable, and you'll be able to directly substitute zero for t and get a finite value for the limit (hint: it's a negative number between -1 and 0).

I hope this helps!

Answer 2

-1/2 = the limit

1/t(sqr(1+t) - 1/t

convert it to one fraction, over a common denominator

= (t - (sqr(1+t))/tsqr(1+t)

try L'Hopital's rule

take derivatives, separately of numerator and denominator

(1 -(1/2)/sqr(1+t))/(t/2)/(t/2)(sqr(1+t) +sqr(1+t))

plug in t=0

= -1/2

if you got something undefined, apply the rule again, and again until

you get a solution, until you get a fraction where you can plug in 0 and not

get an undefined value)

answer = -1/2

a couple strategies

graph it with a graphing calculator and see what happens as t approaches zero

the curve is log shaped

which approaches 0 as t approaches infinity

the curve has a y intercept of (0, -1/2) it =-1/2 at t=0,

it's not "just" a limit that it approaches but never reaches

there's no discontinuity at t=0

or

rationalize the numerator

most people rationalize the denominator