Evaluate? lim (5/(t√(1+t))-5/t)‎ t→0‎

The answer must be -2.5, but I'm stuck because we have 5 which is a big number...plz explain when you solve it!

1 Answer
Timber Lin
Feb 13, 2018

#lim_(trarr0)(5/(tsqrt(1+t))-5/t)#
plugging in 0 directly gives #oo-oo#, so more simplification is needed.

combine the fractions:
#lim_(trarr0)(5/(tsqrt(1+t))-(5sqrt(1+t))/(tsqrt(1+t)))#
#lim_(trarr0)((5-5sqrt(1+t))/(tsqrt(1+t)))#

conjugate the numerator:
#lim_(trarr0)(((5-5sqrt(1+t))(5+5sqrt(1+t)))/(tsqrt(1+t)(5+5sqrt(1+t))))#
#lim_(trarr0)((25-25(1+t))/(tsqrt(1+t)(5+5sqrt(1+t))))#
#lim_(trarr0)((25-25-25t)/(tsqrt(1+t)(5+5sqrt(1+t))))#
#lim_(trarr0)((-25t)/(tsqrt(1+t)(5+5sqrt(1+t))))#

divide t from the numerator and denominator:
#lim_(trarr0)((-25)/(sqrt(1+t)(5+5sqrt(1+t))))#

now plug in #t=0#
#lim_(trarr0)(-25/(sqrt(1+0)(5+5sqrt(1+0))))#
#lim_(trarr0)(-25/(sqrt(1)(5+5sqrt(1))))#
#lim_(trarr0)(-25/(5+5))#
#lim_(trarr0)(-25/10)#
#lim_(trarr0)(-5/2)#
#=-2.5#

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