QuestionFind the limit: $\lim _{x \rightarrow 0}\left(\frac{e^{3 x}-1}{x}\right)$ .

Studdy Solution

STEP 1

Assumptions1. We are given the limit expression $\lim{x \rightarrow0}\left(\frac{e^{3 x}-1}{x}\right)$ . We know that the limit of a function as x approaches a certain value is the value that the function approaches as x gets closer and closer to that value.
3. We also know the limit property $\lim{x \rightarrow0}\left(\frac{e^{x}-1}{x}\right) =1$ .

STEP 2

We can rewrite the given limit expression to use the known limit property. We do this by factoring out a from the exponent in the numerator.
$\lim{x \rightarrow0}\left(\frac{e^{ x}-1}{x}\right) = \lim{x \rightarrow0}\left(\frac{e^{ x}-1}{x} \times\right)$

STEP 3

Now, we can apply the limit property $\lim{x \rightarrow0}\left(\frac{e^{x}-1}{x}\right) =1$ to the expression inside the limit.
$\lim{x \rightarrow0}\left(\frac{e^{3 x}-1}{3x} \times3\right) = \lim{x \rightarrow0}\left(\frac{e^{3 x}-1}{3x}\right) \times3 =1 \times3$

STEP 4

Finally, we calculate the value of the limit.
$1 \times3 =3$ So, $\lim{x \rightarrow0}\left(\frac{e^{3 x}-1}{x}\right) =3$ .

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