limit x->+inf (2*x+1)/x — step-by-step solution

Let's evaluate the limit of

\frac{2 x + 1}{x}

when

x \to +∞

ⓘ A quantity that grows without bound multiplied by a quantity not tending to 0 still grows without bound, the signs giving +∞ or -∞

The limit of

2 x

+∞

when

x \to +∞

ⓘ A quantity that grows without bound plus a finite or bounded quantity still grows without bound

The limit of

2 x + 1

+∞

when

x \to +∞

For

\frac{2 x + 1}{x}

, we encounter the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

To find the limit, we can factor out the dominant term and transform

\frac{2 x + 1}{x}

into

2 + \frac{1}{x}

ⓘ A number divided by a quantity that grows without bound tends to 0

The limit of

\frac{1}{x}

0

⁺ when

x \to +∞

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

2 + \frac{1}{x}

2

when

x \to +∞

So the limit of

\frac{2 x + 1}{x}

2

when

x \to +∞

▶Limit:

2