The functions in exponential notation are appeared in limits and we require some special formulas to find the limits of expressions in which the exponential form functions are involved. So, let’s learn some standard limit formulas with proofs and examples to learn how to find the limits of exponential functions.

$(1).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^{\displaystyle \normalsize x}-1}{x}} \normalsize \,=\, 1$

The limit of mathematical constant $e$ raised to the power of $x$ minus $1$ divided by $x$ as $x$ approaches $0$ is equal to one.

$(2).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}} \normalsize \,=\, \log_{e}{a}$

The limit of a constant $a$ raised to the power of $x$ minus $1$ divided by $x$ as $x$ approaches $0$ is equal to the natural logarithm of constant $a$.

$\displaystyle \large \lim_{x\,\to\,a}{\normalsize \dfrac{x^{\displaystyle \normalsize n}-a^{\displaystyle \normalsize n}}{x-a}} \normalsize \,=\, n.a^{{\displaystyle n}\,-\,\large 1}$

The limit of $x$ raised to the power $n$ minus $a$ raised to the power $n$ divided by $x$ minus $a$ is equal to the $n$ times $a$ raised to the power of $n$ minus one, when the value of $x$ approaches $a$.

$(1).\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize (1+x)^{\Large \frac{1}{x}}} \normalsize \,=\, e$

The limit of $1$ plus $x$ whole raised to the power of the reciprocal of $x$ as $x$ tends to zero is equal to mathematical constant $e$.

$(2).\,\,$ $\displaystyle \large \lim_{x\,\to\,\infty}{\normalsize \bigg(1+\dfrac{1}{x}\bigg)^{\displaystyle x}} \normalsize \,=\, e$

The limit of $1$ plus the reciprocal of $x$ whole raised to the power of $x$ as $x$ approaches to infinity is equal to Napier’s constant $e$.

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