lim
t→0
√a+t-√a-t
t
for a > 0
To evaluate the limit of the expression (√(a+t) - √(a-t))/t as t approaches 0, we can use the following approach:
1. Start by rationalizing the numerator:
Multiply both the numerator and denominator by the conjugate of the numerator, which is (√(a+t) + √(a-t)).
[(√(a+t) - √(a-t))/(t)] * [((√(a+t) + √(a-t)) / (√(a+t) + √(a-t))]
2. Simplify the numerator:
(√(a+t))^2 - (√(a-t))^2
(a+t) - (a-t)
2t
3. Simplify the denominator:
t
4. Now, simplify the expression:
(2t)/(t)
2
Therefore, the limit of (√(a+t) - √(a-t))/t as t approaches 0 is 2.
To evaluate the limit as t approaches 0 of the expression √(a+t) - √(a-t) / t, we can start by rationalizing the expression.
Step 1: Rationalize the expression
Since the numerator contains a difference of square roots, we can multiply the expression by the conjugate of the denominator, which is √(a+t) + √(a-t).
√(a+t) - √(a-t) / t * (√(a+t) + √(a-t)) / (√(a+t) + √(a-t))
Step 2: Simplify the expression
Using the property (a - b)(a + b) = a^2 - b^2, we can simplify the expression as follows:
(√(a+t))^2 - (√(a-t))^2 / t * (√(a+t) + √(a-t)) / (√(a+t) + √(a-t))
(a + t) - (a - t) / t * (√(a+t) + √(a-t)) / (√(a+t) + √(a-t))
2t / t * (√(a+t) + √(a-t)) / (√(a+t) + √(a-t))
Step 3: Cancel out the common terms
Cancelling out the t terms, we get:
2 / (√(a+t) + √(a-t))
Step 4: Evaluate the limit
Now we can evaluate the limit as t approaches 0:
lim(t→0) 2 / (√(a+t) + √(a-t))
As t approaches 0, both (a+t) and (a-t) approach a. Therefore, we have:
lim(t→0) 2 / (√(a+a) + √(a-a))
lim(t→0) 2 / (√(2a) + √0)
lim(t→0) 2 / (√(2a))
Therefore, the limit as t approaches 0 of √(a+t) - √(a-t) / t for a > 0 is equal to 2 / (√(2a)).
To find the limit of the given function as t approaches 0, we can use the concept of the difference of squares.
1. Start with the given function: √(a + t) - √(a - t) / t
2. Multiply the numerator and denominator by the conjugate of the numerator: (√(a + t) - √(a - t)) * (√(a + t) + √(a - t)) / t * (√(a + t) + √(a - t))
3. Apply the difference of squares formula: (a + t) - (a - t) / (t * (√(a + t) + √(a - t)))
4. Simplify the numerator: a + t - a + t / (t * (√(a + t) + √(a - t)))
5. Cancel out like terms in the numerator: 2t / (t * (√(a + t) + √(a - t)))
6. Simplify the expression: 2 / (√(a + t) + √(a - t))
To find the limit as t approaches 0, we substitute t = 0 into the simplified expression:
lim(t→0) 2 / (√(a + 0) + √(a - 0))
lim(t→0) 2 / (√a + √a)
lim(t→0) 2 / 2√a
lim(t→0) 1 / √a
Therefore, the limit of the function as t approaches 0 for a > 0 is 1 / √a.