lim((2*exp(x)+x)/(exp(x)+1), x->+inf)

1) Let's evaluate the limit of

\frac{2 \exp (x) + x}{\exp (x) + 1}

when

x \to +∞

The limit of

\exp (x)

+∞

when

x \to +∞

The limit of

2 \exp (x)

+∞

when

x \to +∞

The limit of

2 \exp (x) + x

+∞

when

x \to +∞

The limit of

\exp (x) + 1

+∞

when

x \to +∞

For

\frac{2 \exp (x) + x}{\exp (x) + 1}

, we encounter the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

2) To find the limit, we divide the numerator and the denominator by

\exp (x)

Let's reduce:

\frac{2 \exp (x) + x}{\exp (x)}

2 + \frac{x}{\exp (x)}

3) Let's reduce:

\frac{\exp (x) + 1}{\exp (x)}

1 + \frac{1}{\exp (x)}

We obtain a new fraction whose limit can be determined:

\frac{2 + \frac{x}{\exp (x)}}{1 + \frac{1}{\exp (x)}}

For

\frac{x}{\exp (x)}

, we encounter the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

By comparative growth,

\exp (x)

grows faster than

x

x \to +∞

The limit of

\frac{x}{\exp (x)}

0

⁺ when

x \to +∞

The limit of

2 + \frac{x}{\exp (x)}

2

when

x \to +∞

The limit of

\frac{1}{\exp (x)}

0

⁺ when

x \to +∞

The limit of

1 + \frac{1}{\exp (x)}

1

when

x \to +∞

The limit of

\frac{2 + \frac{x}{\exp (x)}}{1 + \frac{1}{\exp (x)}}

2

when

x \to +∞

So the limit of

\frac{2 \exp (x) + x}{\exp (x) + 1}

2

when

x \to +∞

▷Answer: $2$