evaluate (2/tsqrt(4+t)-1/t))

Answer 1

I will assume the problem is:

Rationalize the left term by multiplying by √(4+t)/√(4+t) which makes the left term:

(2√(4+t))/(t(4+t))

Get a common denominator to combine the left and right terms:

(2√(4+t))/(t(4+t)) - (1•(4+t))/(t•(4+t))

(2√(4+t) - (4+t))/(t(4+t))

Multiply top and bottom by the conjugate of the numerator "(2√(4+t) + (4+t))":

(4(4+t) - (4+t)²)/(t(4 + t)(2√(4+t) + (4+t)))

(16 + 4t - 16 - 8t - t²)/(t(4 + t)(2√(4+t) + (4+t)))

(-4t - t²)/(t(4 + t)(2√(4+t) + (4+t)))

(t(-4 - t))/(t(4 + t)(2√(4+t) + (4+t)))

Cancel "t" on top and bottom:

(-4 - t)/((4 + t)(2√(4+t) + (4+t)))

Plug in zero:

-4/((4)(2•2 + 4))

-4/32

-1/8

So the limit is -1/8