The limit as x approaches 0 of a times (2 + e^(1/x))/(1 + e^(1/x)) plus (1 + |x|)^(1/x) exists. Find the value of a.

Step-by-step solution

1. 1

   Recognise why the limit is two-sided trouble. Both $e^{1/x}$ and $|x|$ behave differently on the two sides of $0$, so the limit exists precisely when the right-hand and left-hand limits are equal. Compute each side separately and then force them to match.

2. 2

   Right-hand side: $x\to 0^{+}$. Here $\frac1x\to+\infty$, so $e^{1/x}\to+\infty$ and $\frac{2+e^{1/x}}{1+e^{1/x}}\to 1$ (divide top and bottom by $e^{1/x}$). Also $|x|=x$, so the second term is $(1+x)^{1/x}\to e$. Hence $L_{+}=a+e$.

3. 3

   Left-hand side: $x\to 0^{-}$. Now $\frac1x\to-\infty$, so $e^{1/x}\to 0$ and $\frac{2+e^{1/x}}{1+e^{1/x}}\to\frac{2}{1}=2.$ Also $|x|=-x$, so the second term is $(1-x)^{1/x}$. Putting $t=-x\to 0^{+}$ gives $(1+t)^{-1/t}\to e^{-1}$. Hence $L_{-}=2a+\dfrac1e$.

4. 4

   Set the two one-sided limits equal. $a+e=2a+\frac1e\ \Longrightarrow\ a=e-\frac1e.$

5. 5

   Check the common value. With $a=e-\tfrac1e\approx 2.35040$, the right limit is $a+e\approx 5.06868$ and the left limit is $2a+\tfrac1e\approx 4.70081+0.36788\approx 5.06868$. The two agree, so the limit does exist for this $a$ and for no other.