I'm reviewing for a test tomorrow and two of the questions on the review had me a little stumped.
Lim Sq. Rt.(x+5) - Sq. Rt (5) all divided by
X-->0 X
I was able to rationalize and ended up with
(X)/ x ((sq. rt.(x+5))+ sq. rt.(5)) which gives me
1/ sq. rt.(5)+ sq. rt.(5) Now I'm stuck as to how to simplify it. I know I can input it in the calculator, but the review has the answer in fraction form.
The answer given is Sq. Rt.(5) / 10. How do I get it in that form?
lim (√(x+5)-√5)/x
use L'Hospital's Rule:
lim [1/2√(x+5)]/1 = 1 / 2√5
To simplify the expression further, you can rationalize the denominator.
Starting with your expression: 1/ (sqrt(5) + sqrt(5))
To rationalize the denominator, we can multiply the expression by its conjugate, which is (sqrt(5) - sqrt(5)).
Multiply the numerator and denominator by (sqrt(5) - sqrt(5)):
(1/(sqrt(5) + sqrt(5))) * (sqrt(5) - sqrt(5)) / (sqrt(5) - sqrt(5))
Now simplify the numerator:
(1 * sqrt(5) - 1 * sqrt(5)) / (sqrt(5) - sqrt(5))
The terms with sqrt(5) cancel out:
0 / (sqrt(5) - sqrt(5))
Since anything divided by zero is undefined, the simplified expression is equal to zero.
Therefore, the given answer, sqrt(5)/10, is not correct.
To simplify the expression, you need to rationalize the denominator further. Here's how you can do it step by step:
1. Start with the expression: 1 / (√(5) + √(x+5))
2. Multiply both the numerator and denominator by the conjugate of the denominator, which is (√(5) - √(x+5)). By multiplying the conjugate, we eliminate the square root in the denominator.
(1 / (√(5) + √(x+5))) * ((√(5) - √(x+5)) / (√(5) - √(x+5)))
Simplifying this, we get:
(√(5) - √(x+5)) / ((√(5))^2 - (√(x+5))^2)
(√(5) - √(x+5)) / (5 - (x+5))
(√(5) - √(x+5)) / (5 - x - 5)
(√(5) - √(x+5)) / (-x)
3. Simplify the expression further by factoring a negative out of the denominator:
(√(5) - √(x+5)) / (-1 * x)
-(√(5) - √(x+5)) / x
4. Finally, rearrange the expression to match the desired form:
-((√(x+5) - √(5)) / x)
Therefore, the simplified expression is -((√(x+5) - √(5)) / x). Note that the negative sign can also be brought into the numerator, but it would not affect the final answer when canceling out with the negative sign in the denominator.
Keep in mind that there may be different forms or variations of the expression, but the above steps illustrate one way to simplify it.