Limits of Exponential Functions

The limit of the exponential function is an important formula in limit theory and is frequently used in limit computation.

Basic Form

Detailed Derivation

Step 1: Change of variable

• Let $t = e^x - 1$
• Then $e^x = 1 + t$
• Taking logarithms on both sides: $x = \ln(1 + t)$

Step 2: Limit conversion As $x \to 0$:

• Since $t = e^x - 1$, as $x \to 0$, $e^x \to 1$, so $t \to 0$

Step 3: Establish the equality $\lim_{x \to 0} \frac{e^x - 1}{x} = \lim_{t \to 0} \frac{t}{\ln(1 + t)}$

Step 4: Use the second important limit The second important limit: $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

Detailed explanation of the second important limit:

The original form of the second important limit is: $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

Through a change of variable, we get the reciprocal form: $\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$

Taking logarithms on both sides: $\lim_{x \to 0} \frac{1}{x} \ln(1 + x) = 1$

That is: $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$

Using the inverse relationship between the exponential and logarithmic functions, we finally get: $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

Therefore: $\lim_{t \to 0} \frac{t}{\ln(1 + t)} = 1$

Key insight: The core idea of this equality is that the exponential function and the logarithmic function are inverse operations. Through a change of variable, the limit of the exponential function is transformed into the limit of the logarithmic function, and the result is finally obtained using the known second important limit.

Generalized Form

$\lim_{x \to 0} \frac{e^{kx} - 1}{x} = k$

Equivalent Infinitesimal

Using this important limit, we obtain an important equivalent infinitesimal:

• As $x \to 0$, $e^x - 1 \sim x$

Worked Examples

Example 1

Find $\lim_{x \to 0} \frac{e^{2x} - 1}{x}$

Solution: $\lim_{x \to 0} \frac{e^{2x} - 1}{x} = 2$

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Practice Problems

Exercise 1

Find the limit $\lim_{x \to 0} \frac{e^{3x} - 1}{x}$.

Exercise 2

Find the limit $\lim_{x \to 0} \frac{e^{5x} - 1}{2x}$.

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Summary

Symbols Used in This Article