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• Calculus

Power Law of Limit

Formula

$\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{g{(x)}}}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{\, \displaystyle \large \lim_{x \,\to\, a} \, {\normalsize g{(x)}}}}$

It is a property of power rule, used to find the limit of an exponential function whose base and exponent are in a function form.

Proof

$x$ is a variable and two functions $f{(x)}$ and $g{(x)}$ are defined in terms of $x$. The limits of $f{(x)}$ and $g{(x)}$ as $x$ closer to $a$ are written mathematically in calculus as follows.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $\,=\,$ $f{(a)}$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$ $\,=\,$ $g{(a)}$

Limit of Functions in Exponential form

Assume, the functions $f{(x)}$ and $g{(x)}$ are formed a function in exponential form.

${f{(x)}}^{g{(x)}}$

Now, find the limit of this exponential function as $x$ approaches $a$.

$\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{g{(x)}}}$

Find the limit of the function

Find the limit of the exponential function by substituting $x$ by $a$.

$= \,\,\, {f{(a)}}^{g{(a)}}$

Get Power Rule of Limit

The limits of functions $f{(x)}$ and $g{(x)}$ as $x$ tends to $a$ are $f{(a)}$ and $g{(a)}$ respectively. Therefore, it can be written that $f{(a)}$ and $g{(a)}$ as the limits of functions $f{(x)}$ and $g{(x)}$ respectively.

$\implies {f{(a)}}^{g{(a)}}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{\, \displaystyle \large \lim_{x \,\to\, a} \, {\normalsize g{(x)}}}}$

Actually, the value of $f{(a)}$ is raised to the power of $g{(a)}$ is determined as the limit of the $f{(x)}$ is raised to the power of $g{(x)}$ as $x$ closer to $a$.

$\,\,\, \therefore \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{g{(x)}}}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a} \, {\normalsize {f{(x)}}^{\, \displaystyle \large \lim_{x \,\to\, a} \, {\normalsize g{(x)}}}}$

Therefore, the limit property is proved that the limit of $f{(x)}$ is raised to the power of $g{(x)}$ as $x$ approaches $a$ equals to the limit of $f{(x)}$ as $x$ approaches $a$ is raised to the power of the limit of $g{(x)}$ as $x$ closer to $a$.

The limit rule is completely in exponential notation. So, it is called as the power rule of limit in calculus.