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

Compute the limit of

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

x \to +∞

The limit of

\ln (x)

+∞

when

x \to +∞

The limit of

\exp (x)

+∞

when

x \to +∞

For

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

, we encounter the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

By comparative growth,

\exp (x)

grows faster than

\ln (x)

x \to +∞

The limit of

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

0

⁺ when

x \to +∞

For

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

, the limit takes the indeterminate form

+∞ \cdot 0

when

x

→

+∞

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)}

, the limit takes the indeterminate form

\frac{+∞}{+∞}

when

x

→

+∞

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)}}

, the limit takes the indeterminate form

\frac{+∞}{+∞}

when

x

→

+∞

By comparative growth,

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

grows faster than

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 +∞

▷Result: $0$