Solve ;limit as x approaches zero (4-2x^2)1/x
To solve this limit, we can use logarithmic differentiation.
First, we rewrite the expression as:
(4-2x^2)^(1/x) = e^(ln(4-2x^2) / x)
Then, using properties of logarithms, we can rewrite the expression inside the natural logarithm:
ln(4-2x^2) = ln(2(2-x)(2+x))
= ln(2) + ln(2-x) + ln(2+x)
Now, we can take the natural logarithm of the entire original expression:
ln((4-2x^2)^(1/x)) = ln(e^(ln(4-2x^2) / x))
= ln(4-2x^2) / x
= (ln(2) + ln(2-x) + ln(2+x)) / x
As x approaches zero, the denominator of this expression approaches zero, so we have an indeterminate form of infinity over infinity. We can use L'Hopital's rule to evaluate this limit:
lim x->0 (ln(2) + ln(2-x) + ln(2+x)) / x
= lim x->0 ((1/(2-x)) + (-1/(2+x)))
= 1/2 - 1/2
= 0
Therefore, the limit as x approaches zero of (4-2x^2)^(1/x) is e^0, which simplifies to 1.
Answer: 1.
To solve the limit as x approaches zero of (4-2x^2)^(1/x), we can use the properties of logarithms and limits.
Step 1: Take the natural logarithm (ln) of both sides of the expression to simplify it. Recall that ln(a^b) = b*ln(a).
ln[(4-2x^2)^(1/x)]
Step 2: Apply the limit to the natural logarithm of the expression. The limit of ln(f(x)) as x approaches a is equal to ln(L), where L is the limit of f(x) as x approaches a.
lim(x->0) ln[(4-2x^2)^(1/x)]
Step 3: Use the properties of logarithms to rewrite the expression as a product.
lim(x->0) [ln(4-2x^2)]*(1/x)
Step 4: Evaluate the limits of each term separately.
lim(x->0) ln(4-2x^2) * lim(x->0) (1/x)
Step 5: Simplify the first term limit. Recall that ln(e) = 1, where e is the Euler's number.
ln(4-2(0)^2) = ln(4) = ln(2^2) = 2ln(2)
Step 6: Simplify the second term limit. Recall that the limit of 1/x as x approaches 0 is positive or negative infinity (-∞ or +∞).
lim(x->0) (1/x) = ±∞
Step 7: Combine the results. Since one term approaches infinity and the other is a finite value, the overall limit does not exist.
Therefore, the limit of (4-2x^2)^(1/x) as x approaches zero does not exist.