limit x->+inf (3*x^2-1)/(x^2+5) — step-by-step solution

1) Compute the limit of

\frac{3 x^{2} - 1}{x^{2} + 5}

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

The limit of

3 x^{2}

+∞

when

x \to +∞

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

The limit of

3 x^{2} - 1

+∞

when

x \to +∞

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

The limit of

x^{2} + 5

+∞

when

x \to +∞

For

\frac{3 x^{2} - 1}{x^{2} + 5}

, the limit takes the indeterminate form

\frac{+∞}{+∞}

when

x

→

+∞

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

\frac{3 x^{2} - 1}{x^{2} + 5}

into

\frac{3 - \frac{1}{x^{2}}}{1 + \frac{5}{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

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

3

when

x \to +∞

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

The limit of

\frac{5}{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{5}{x^{2}}

1

when

x \to +∞

The limit of

\frac{3 - \frac{1}{x^{2}}}{1 + \frac{5}{x^{2}}}

3

when

x \to +∞

So the limit of

\frac{3 x^{2} - 1}{x^{2} + 5}

3

when

x \to +∞

▶Limit:

3