Limits of Exponential and Logarithmic Functions

Direct substitution. The exponential function $e^x$ (base

$e\approx2.71828$) is defined and continuous for every real $x$, so

by direct substitution, exactly as for polynomials and trigonometric functions. Similarly, the

natural logarithm $\ln x$ is continuous throughout its domain $x>0$, so $\lim_{x\to a}\ln x=\ln a$

for any $a>0$.

A standard limit (stated, not re-derived here). As with $\sin x/x$ in Section 4, the

exponential analogue is stated and used here as a previously-established standard result:

A companion result for the natural logarithm, obtained via the substitution $y=e^x-1$ (so

$x=\ln(1+y)$, and $y\to0$ exactly when $x\to0$), is

General exponential base $a>0,\ a\neq1$. Writing $a^x=e^{x\ln a}$ (the defining relation

between any positive base and the natural exponential), the limit $\lim_{x\to0}(a^x-1)/x$ can be

derived from the $e^x$ standard limit above, rather than proved separately from scratch. Set

$u=x\ln a$; as $x\to0$, also $u\to0$ (assuming $a\neq1$, so $\ln a\neq0$), and

As $x\to0$ (hence $u\to0$), $\dfrac{e^u-1}{u}\to1$ by the standard limit, so

This derived result specialises back to $\lim_{x\to0}(e^x-1)/x=1$ when $a=e$ (since $\ln e=1$),

confirming consistency.

Scaling trick, as in Section 4. For any nonzero constant $k$,

by the same substitution-scaling method used for $\sin(kx)/x$. …