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Step-by-Step Solution
Step 1: Understand the Given Limit
We have the limit:
\lim_{x \to 0} \frac{ax - (e^{4x} - 1)}{ax\,(e^{4x} - 1)},
which is given to exist and is equal to b . Our goal is to find the value of a - 2b.
Step 2: Determine the Condition for the Limit to Exist
First, let us check what happens if we directly substitute x = 0 . We see that:
The numerator ax - (e^{4x} - 1) becomes a \cdot 0 - (1 - 1) = 0.
The denominator ax\,(e^{4x} - 1) also becomes 0 \cdot (1 - 1) = 0.
Hence, the limit is in an indeterminate form 0/0 , suggesting we can use L’Hôpital’s Rule or series expansion to simplify.
Step 3: Check for the Existence of the Limit Using an Expansion
A quicker way to see if the limit will exist is by comparing the leading terms in the numerator and denominator. Let us expand e^{4x} near x = 0.
Recall the series expansion:
e^{4x} = 1 + 4x + 8x^2 + \cdots
Then:
Numerator:
ax - (e^{4x} - 1) = ax - \bigl((1 + 4x + \cdots) - 1\bigr) = ax - (4x + \cdots) = (a-4)x + \cdots
Denominator:
ax \,(e^{4x} - 1) = ax \bigl(4x + \cdots\bigr) = 4a\,x^2 + \cdots
If a \neq 4, the fraction will behave like
\frac{(a-4)x}{4a\,x^2} = \frac{a-4}{4a} \cdot \frac{1}{x},
which goes to infinity as x \to 0 and thus the limit would not exist. Therefore, for the limit to exist, we must have:
a - 4 = 0 \quad \Longrightarrow \quad a = 4.
Step 4: Substitute a = 4 Back into the Limit
Now that a = 4, the expression becomes:
\lim_{x \to 0} \frac{4x - (e^{4x} - 1)}{4x\,(e^{4x} - 1)}.
We can either:
Use L'Hôpital's Rule again.
Use a series expansion for the refined form.
Let us do a quick series expansion again to see the behavior:
Numerator (with a = 4 ):
4x - \bigl(e^{4x} - 1\bigr)
= 4x - \bigl(1 + 4x + 8x^2 + \cdots - 1\bigr)
= 4x - 4x - 8x^2 + \cdots = -8x^2 + \cdots
Denominator:
4x \,(e^{4x} - 1)
= 4x \,\bigl(4x + 8x^2 + \cdots\bigr)
= 16x^2 + \cdots
Hence,
\lim_{x \to 0} \frac{-8x^2 + \cdots}{16x^2 + \cdots}
= \frac{-8}{16} = -\frac{1}{2}.
Therefore, b = -\frac{1}{2}.
Step 5: Final Computation of a - 2b
We found a = 4 and b = -\tfrac{1}{2} . So:
a - 2b
= 4 - 2\left(-\frac{1}{2}\right)
= 4 + 1
= 5.
Thus, the value of a - 2b is \boxed{5} .
