Home > Calculus > Determining Limits Algebraically

How do you evaluate the limit #(sqrt(x+1)-2)/(x-3)# as x approaches #3#?

Answer 1

Aurora Alvarado

#= 1/4#

#lim_(x to 3) (sqrt{(x+1)}-2)/(x-3)#

with #x = 3 + h, h = x-3, 0 < abs(h) "<<" 1#

becomes #lim_(h to 0) (sqrt{3 + h +1}-2)/(3+ h-3)#

#lim_(h to 0) 1/h * (sqrt{4 + h}-2)#

#lim_(h to 0) 1/h *( 2 sqrt{1 + h/4}-2)#

by Taylor expansion

#lim_(h to 0) 1/h * ( 2 (1 + 1/2 * h/4 + mathcal(O)(h^2)) -2 )#

#lim_(h to 0) 1/h * 2 ( 1/2 * h/4 + mathcal(O)(h^2)) #

#lim_(h to 0) 1/h * ( h/4 + mathcal(O)(h^2)) #

#lim_(h to 0) 1/4 + mathcal(O)(h) = 1/4#

You can see clearly that the limit is the same as x approaches 3 on both sides as the sign of h is in the error term.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Luke Alvarez

To evaluate the limit of (sqrt(x+1)-2)/(x-3) as x approaches 3, we can use algebraic manipulation. By multiplying both the numerator and denominator by the conjugate of the numerator, which is sqrt(x+1)+2, we can simplify the expression. After simplification, we find that the limit is equal to 1/4.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.