limit x->infinity x*ln(x)/exp(x)

1) Let's evaluate the limit of

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

when

x \to +∞

ⓘ This is a reference limit of the function ln

The limit of

\ln (x)

+∞

when

x \to +∞

ⓘ This is a reference limit of the function exp

The limit of

\exp (x)

+∞

when

x \to +∞

For

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

, we encounter the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

Comparative growth shows that

\exp (x)

dominates

\ln (x)

x \to +∞

The limit of

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

0

⁺ when

x \to +∞

For

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

, we obtain the indeterminate form

+∞ \cdot 0

x

tends to

+∞

2) To resolve the indeterminate form 0 × ∞, we rewrite the product

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

as the quotient

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

For

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

, we obtain the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

By comparative growth,

\exp (x)

grows faster than

\ln (x)

x \to +∞

The limit of

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

+∞

when

x \to +∞

For

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

, we obtain the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

Comparative growth shows that

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

dominates

x

x \to +∞

The limit of

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

0

⁺ when

x \to +∞

So the limit of

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

0

⁺ when

x \to +∞

▶Limit:

0