Evaluate the limit as x approaches infinity of ((x + 3)/(x - 1)) raised to the power (x + 3).

Step-by-step solution

1. 1

   Identify the form. The base tends to $1$ (numerator and denominator both grow like $x$) while the exponent tends to $\infty$. That is the indeterminate form $1^{\infty}$ — it cannot be evaluated by substituting the two limits separately.

2. 2

   Rewrite the base as 1 plus a small piece. Divide out the denominator:

   $\frac{x+3}{x-1}=\frac{(x-1)+4}{x-1}=1+\frac{4}{x-1}.$

   This form is what lets you match the standard limit $\left(1+\tfrac1u\right)^{u}\to e$.

3. 3

   Substitute u = x - 1 to make the pattern exact. As $x\to\infty$, $u\to\infty$ and the exponent $x+3=u+4$:

   $\left(1+\frac{4}{u}\right)^{u+4}=\left[\left(1+\frac{4}{u}\right)^{u}\right]\cdot\left(1+\frac{4}{u}\right)^{4}.$

4. 4

   Take the two limits separately. The bracket is the standard limit with $a=4$:

   $\left(1+\frac{4}{u}\right)^{u}\longrightarrow e^{4},$

   while the trailing factor $\left(1+\tfrac4u\right)^{4}\to 1^{4}=1$. The extra $+3$ in the original exponent therefore has no effect on the answer — only the leading $x$ matters.

5. 5

   Combine.

   $\lim_{x\to\infty}\left(\frac{x+3}{x-1}\right)^{x+3}=e^{4}\cdot 1=e^{4}\approx 54.598.$

6. 6

   Confirm numerically. At $x=10^{6}$ the expression evaluates to $54.59859$, against $e^{4}=54.59815$ — agreeing to four significant figures, with the small gap shrinking like $1/x$ as expected.