Find the value of limit #{sqrt(1+x^2)-sqrt(1+x)}/{sqrt(1+x^3)-sqrt(1+x)}# as x approaches to 0?
2 Answers
Explanation:
Remove the subtraction of radicals by multiplying numerator and denominator by
(That is the product of the conjugates of the numerator and the denominator.)
We get
# = (sqrt(1+x^3) + sqrt(1+x))/((x+1)(sqrt(1+x^2) + sqrt(1+x))#
Evaluating the limit as
# = (sqrt(1+0)+sqrt(1+0))/((0+1)(sqrt(1+0)+sqrt(1+0))) = 2/2=1#
As a check, here's the graph of the function
graph{(sqrt(1+x^2)-sqrt(1+x))/(sqrt(1+x^3)-sqrt(1+x)) [-8.794, 8.98, -1.626, 7.264]}
Explanation:
Plugging in zero gives:
Using L'Hospital's Rule:
Differentiate numerator and denominator.
Numerator:
Denominator:
Plugging in zero: