limx->0 (3^x-6^x)/x please help here do not really know where to even start
see earlier post
woops sorry didn't see someone had already posted this
To evaluate the limit limx->0 (3^x - 6^x)/x, we can use a combination of algebraic manipulation and the properties of limits:
Step 1: Simplify the expression
We can rewrite the expression (3^x - 6^x)/x as [(3^x - 6^x)]/x.
Step 2: Apply the identity (a^x - b^x) = (a - b)(a^(x-1) + a^(x-2)b + a^(x-3)b^2 + ... + ab^(x-2) + b^(x-1)).
In this case, a = 3, b = 6, and x is approaching 0. Using this identity, we can rewrite the expression as [(3 - 6)(3^(x-1) + 3^(x-2)6 + 3^(x-3)(6^2) + ... + 3(6^(x-2)) + 6^(x-1))]/x.
Step 3: Simplify further
We now have [(3 - 6)(3^(x-1) + 3^(x-2)6 + 3^(x-3)(6^2) + ... + 3(6^(x-2)) + 6^(x-1))]/x, which simplifies to (-3)(3^(x-1) + 3^(x-2)6 + 3^(x-3)(6^2) + ... + 3(6^(x-2)) + 6^(x-1))/x.
Step 4: Evaluate the limit
To evaluate the limit limx->0 (-3)(3^(x-1) + 3^(x-2)6 + 3^(x-3)(6^2) + ... + 3(6^(x-2)) + 6^(x-1))/x, we can first factor out the constant (-3) and then substitute x = 0.
Taking the limit as x approaches 0, (-3)(3^(0-1) + 3^(0-2)6 + 3^(0-3)(6^2) + ... + 3(6^(0-2)) + 6^(0-1))/0 reduces to (-3)(3^(-1) + 3^(-2)6 + 3^(-3)(6^2) + ... + 3(6^(-2)) + 6^(-1))/0.
Next, we can simplify the expression under the limit (-3)(3^(-1) + 3^(-2)6 + 3^(-3)(6^2) + ... + 3(6^(-2)) + 6^(-1)) as (-3)(1/3 + 1/(3*6) + 1/(3*6^2) + ... + 1/(3*6^(n-1)) + 1/6).
Finally, substituting x = 0 into the expression, we get (-3)(1/3 + 1/(3*6) + 1/(3*6^2) + ... + 1/(3*6^(n-1)) + 1/6)/0.
However, since we have 0 in the denominator, the expression is undefined and the limit does not exist.