How do you evaluate the limit #(sqrt(x+6)-x)/(x-3)# as x approaches #3#?

Answer 1

#-5/6#

As the function is indeterminate #0/0# when x = 3

Multiply the numerator/denominator by the conjugate of the numerator.

#sqrt(x+6)-x" conjugate "tosqrt(x+6)+x#
#rArr((sqrt(x+6)-x)(sqrt(x+6)+x))/((x-3)(sqrt(x+6)+x)#
#=(x+6-x^2)/((x-3)(sqrt(x+6)+x))#
#=(-cancel((x-3))(x+2))/(cancel((x-3))(sqrt(x+6)+x))#

exclusion x ≠ 3

#=(-(x+2))/(sqrt(x+6)+x)#
#rArrlim_(xto3)(sqrt(x+6)-x)/(x-3)#
#=lim_(xto3)(-(x+2))/(sqrt(x+6)+x)=-5/6#
Sign up to view the whole answer

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

Sign up with email
Answer 2

To evaluate the limit (sqrt(x+6)-x)/(x-3) as x approaches 3, we can use algebraic manipulation. First, we substitute x=3 into the expression:

(sqrt(3+6)-3)/(3-3)

This simplifies to:

(sqrt(9)-3)/0

Since the denominator is 0, we cannot directly evaluate the limit using substitution. However, we can simplify the expression further by rationalizing the numerator:

(sqrt(9)-3)/0 = (3-3)/0 = 0/0

This is an indeterminate form, which means we need to apply additional techniques to evaluate the limit. One approach is to use L'Hôpital's Rule, which states that if we have an indeterminate form of the type 0/0 or ∞/∞, we can differentiate the numerator and denominator separately and then evaluate the limit again.

Differentiating the numerator and denominator, we get:

d/dx (sqrt(x+6)-x) / d/dx (x-3)

Applying the derivative, we have:

(1/2)*(x+6)^(-1/2) - 1 / 1

Now, we substitute x=3 into the derivative expression:

(1/2)*(3+6)^(-1/2) - 1 / 1

Simplifying further:

(1/2)*(9)^(-1/2) - 1 / 1

(1/2)*(1/3) - 1 / 1

1/6 - 1 / 1

1/6 - 6/6

-5/6

Therefore, the limit of (sqrt(x+6)-x)/(x-3) as x approaches 3 is -5/6.

Sign up to view the whole answer

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

Sign up with email
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.

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.

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7