Question

Evaluate: \[ \lim_{x \to \infty} \left( \frac{x+8}{x+1} \right)^{x+5} = \, ? \] 

• A. \(e^4\)
• B. \(e^5\)
• C. \(e^{11}\)
• D. \(e^7\)

Hint:

Whenever you see an expression of the form \[ \left(\frac{x+a}{x+b}\right)^x \] first rewrite the base as \[ 1+\frac{a-b}{x+b} \] and then use the standard exponential limit.

Answer 1

The Correct Option is D

Concept:
Use the standard limit: \[ \lim_{x\to\infty}\left(1+\frac{a}{x}\right)^x=e^a \] So we first rewrite the given expression into a comparable standard form. ip

Step 1: Rewrite the base.
\[ \frac{x+8}{x+1}=1+\frac{7}{x+1} \] So the expression becomes: \[ \left(1+\frac{7}{x+1}\right)^{x+5} \] ip

Step 2: Adjust the exponent.
As \(x\to\infty\), \[ x+5 \sim x+1 \] So, \[ \left(1+\frac{7}{x+1}\right)^{x+5} = \left[\left(1+\frac{7}{x+1}\right)^{x+1}\right]^{\frac{x+5}{x+1}} \] ip

Step 3: Apply the standard limit.
Now, \[ \left(1+\frac{7}{x+1}\right)^{x+1} \to e^7 \] and \[ \frac{x+5}{x+1}\to 1 \] Therefore, \[ \lim_{x\to\infty}\left( \frac{x+8}{x+1} \right)^{x+5}=e^7 \] ip Hence, the correct answer is:
\[ \boxed{(D)\ e^7} \]