how do i get the answer for this limit limit x to 0 (4^2x-1)

without using direct substitution
my work so far is

limit x to 0= (4^2x-1)
= (4^2x/4^-1)

is it correct?

I don't see the problem with this limit question other than your notation.

I will assume you meant:
lim 4^(2x -1) , as x ---> 0

Since you don't want to use substitution, which is always my first step, you could look at the graph of y = 4^(2x-1) and observe what it does at the y-axis.
http://www.wolframalpha.com/input/?i=y+%3D+4%5E(2x-1)+from+x%3D+-.5+to+.5

Notice the y-intercept is .25 or 1/4 and the graph is continuous.

my work:
lim 4^(2x -1) , as x ---> 0
= lim 4^(0-1)
= 4^-1
= 1/4

Your work so far is partially correct, but you made a mistake in the exponent of 4. Let's go through the correct steps to evaluate the limit:

1. Start with the given limit: limit x→0 (4^(2x-1)).
2. To avoid direct substitution, we need to simplify the expression first.
3. Rewrite 4^(2x-1) as the product of two exponents: 4^(2x) * 4^(-1).
4. Now, let's work on simplifying each of these exponents separately.

For the first exponent, 2x, we can use the property of exponentiation that says (a^b)^c = a^(b*c). Applying this, we have 4^(2x) = (2^2)^x = 2^(2x).

For the second exponent, 4^(-1), we can rewrite it as the reciprocal of 4^1, using the property that a^(-b) = 1/(a^b). So, 4^(-1) = 1/(4^1) = 1/4.

Now, we have simplified the expression to 2^(2x) * 1/4.

5. Next, we can rewrite 2^(2x) as (2^2)^x = 4^x.

So, the expression becomes 4^x * 1/4.

6. Now we can take the limit as x approaches 0. Recall that the limit of a constant times a function is equal to the constant times the limit of the function. Applying this rule, we have:

limit x→0 (4^x * 1/4) = 1/4 * limit x→0 (4^x).

7. The final step is to evaluate the remaining limit for x→0. Since the base of 4^x is 4, the limit is simply the value of the function at x=0, which is 4^0 = 1.

Therefore, the final answer is 1/4 * 1 = 1/4.

To summarize:

limit x→0 (4^(2x-1)) = limit x→0 (4^x * 1/4) = 1/4 * limit x→0 (4^x) = 1/4 * 1 = 1/4.

So, the limit is equal to 1/4.