limit x->infinity (x^2+1)/x^2 — step-by-step solution

Let's evaluate the limit of

\frac{x^{2} + 1}{x^{2}}

when

x \to +∞

ⓘ A quantity that grows without bound raised to a positive power still grows without bound, to a negative power it tends to 0

The limit of

x^{2}

+∞

when

x \to +∞

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

The limit of

x^{2} + 1

+∞

when

x \to +∞

For

\frac{x^{2} + 1}{x^{2}}

, the limit takes the indeterminate form

\frac{+∞}{+∞}

when

x

→

+∞

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

\frac{x^{2} + 1}{x^{2}}

into

1 + \frac{1}{x^{2}}

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

The limit of

\frac{1}{x^{2}}

0

⁺ when

x \to +∞

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

The limit of

1 + \frac{1}{x^{2}}

1

when

x \to +∞

So the limit of

\frac{x^{2} + 1}{x^{2}}

1

when

x \to +∞

▶Limit:

1