Exponential and Logarithmic Standard Limits

Two more standard limits — both again of the indeterminate $\tfrac00$ form on direct substitution — are used constantly once exponential and logarithmic functions enter business-mathematics problems (compound growth, continuous interest, and, later in this chapter, differentiation of $e^x$ and $\ln x$):

Both say the same kind of thing as §4's formulae: the expression is undefined at $x=0$ itself ($\tfrac00$), but the function it defines approaches exactly $1$ as $x$ gets arbitrarily close to $0$.

How to apply them to a scaled argument (the most common exam variant): for $\displaystyle\lim_{x\to0}\frac{e^{kx}-1}{x}$, substitute $u=kx$ (so $u\to0$ as $x\to0$, and $x=u/k$):

The same substitution technique gives $\displaystyle\lim_{x\to0}\frac{\ln(1+kx)}{x} = k$.