lim(x tends to 3) (square root(x^2+7) -4) divided by (square root(x^2-8) -1)
Pls help and list the steps. Thx
limit ( √(x^2 + 7) - 4)/( √(x^2 - 8) - 1) , as x --> 3
my first step always is to actually sub in the approach value,
sure enough, I get 0/0
That means the "eventually" my expression should factor.
multiply top and bottom by (√(x^2 - 8) + 1)
I peeked at what Wolfram had to say, and it came out with a nice answer
http://www.wolframalpha.com/input/?i=limit+%28+%E2%88%9A%28x%5E2+%2B+7%29+-+4%29%2F%28+%E2%88%9A%28x%5E2+-+8%29+-+1%29++as+x+--%3E+3
let's try L'Hopital's Rule
( √(x^2 + 7) - 4)/( √(x^2 - 8) - 1) , as x --> 3
= lime ( (1/2)(x^2 + 7)^(-1/2) (2x) )/( (1/2)(x^2 - 8)^(-1/2) (2x) ) as x -->3
= lim √(x^2 - 8)/√(x^2 + 7) as x --->3
= √1/√16
= 1/4
thx so much
To evaluate the limit of the given expression as x approaches 3, we can apply the following steps:
Step 1: Start by substituting the value of x into the expression to see if it results in an indeterminate form, such as 0/0 or ∞/∞. In this case, if we substitute x = 3, we get:
(sqrt(3^2+7) - 4) / (sqrt(3^2-8) - 1)
Step 2: Simplify the expression:
(sqrt(9+7) - 4) / (sqrt(9-8) - 1)
(sqrt(16) - 4) / (sqrt(1) - 1)
(4 - 4) / (1 - 1)
0/0
Step 3: Since we obtained an indeterminate form, we can now employ algebraic techniques to simplify the expression further.
Step 4: Apply a conjugate multiplication to rationalize both the numerator and denominator. In the numerator, multiply by the conjugate of the expression sqrt(x^2+7) - 4, which is sqrt(x^2+7) + 4. Similarly, in the denominator, multiply by the conjugate of the expression sqrt(x^2-8) - 1, which is sqrt(x^2-8) + 1.
Step 5: After applying the conjugate multiplication, the expression becomes:
[(sqrt(x^2+7) - 4)(sqrt(x^2+7) + 4)] / [(sqrt(x^2-8) - 1)(sqrt(x^2-8) + 1)]
Step 6: By multiplying the conjugates, the expression simplifies to:
(x^2+7) - 4^2 / (x^2-8) - 1^2
Step 7: Continuing simplification:
(x^2+7) - 16 / (x^2-8) - 1
Step 8: Combine like terms:
(x^2 - 9) / (x^2 - 9)
Step 9: At this point, we can notice that both the numerator and denominator are the same. Thus, they cancel each other out, leaving us with:
1
Step 10: Since there are no more variables in the expression, we can conclude that the limit of the given expression as x approaches 3 is equal to 1.