Evaluate the following limit lim(x→0) (e^3x - e^2x)/x

Question

Evaluate the following limit \(\lim\limits_{\text x \to0}\cfrac{e^{3\text x}-e^{2\text x}}{\text x} \)

lim(x→0) (e^3x - e^2x)/x

Answer 1

As we need to find \(\lim\limits_{\text x \to0}\cfrac{e^{3\text x}-e^{2\text x}}{\text x} \)

lim(x→0) (e^3x - e^2x)/x

We can directly find the limiting value of a function by putting the value of the variable at which the limiting value is asked if it does not take any indeterminate form (0/0 or ∞/∞ or ∞-∞, .. etc.)

Let Z = \(\lim\limits_{\text x \to0}\cfrac{e^{3\text x}-e^{2\text x}}{\text x} \)\(=\cfrac{e^0-e^0}0=\cfrac{1-1}0\)

\(=\cfrac00\)(indeterminate form)

∴ we need to take steps to remove this form so that we can get a finite value.

TIP: Most of the problems of logarithmic and exponential limits are solved using the formula

  \(\lim\limits_{\text x \to0}\cfrac{a^{\text x-1}}{\text x}\) = log a and \(\lim\limits_{\text x \to0}\cfrac{log(1+\text x)}{\text x}=1\)

This question is a direct application of limits formula of exponential and logarithmic limits.

{Adding and subtracting 1 in numerator}

{using algebra of limits}

To get the form as present in the formula we multiply and divide 3 and 2 into both terms respectively:

∴ Z = 3log e – 2log e= 3 - 2 = 1

{using log e = 1}

Hence,

\(\lim\limits_{\text x \to0}\cfrac{e^{3\text x}-e^{2\text x}}{\text x} \) = 1