63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.


limₕ→₀ (1 + 3h)^{2/h}

Verified step by step guidance

1. Recognize that the limit is of the form \(\lim_{h \to 0} (1 + 3h)^{\frac{2}{h}}\), which resembles an expression that can be related to the exponential function \(e^x\) through limits of the form \((1 + k h)^{\frac{1}{h}}\) as \(h \to 0\).
2. To approximate the limit using a calculator, create a table of values for \(h\) approaching 0 from both positive and negative sides (e.g., \(h = 0.1, 0.01, 0.001, -0.1, -0.01, -0.001\)). For each \(h\), compute the value of \((1 + 3h)^{\frac{2}{h}}\).
3. Observe the trend of the values in the table as \(h\) gets closer to 0 to estimate the limit numerically.
4. To confirm the result analytically using l’Hôpital’s Rule, rewrite the limit in a form suitable for applying the rule by taking the natural logarithm: let \(L = \lim_{h \to 0} (1 + 3h)^{\frac{2}{h}}\), then consider \(\ln L = \lim_{h \to 0} \frac{2}{h} \ln(1 + 3h)\).
5. Apply l’Hôpital’s Rule to the limit \(\lim_{h \to 0} \frac{2 \ln(1 + 3h)}{h}\) by differentiating numerator and denominator with respect to \(h\), then exponentiate the result to find the original limit.

Key Concepts

Limit of a Function as a Variable Approaches a Value

The limit describes the value that a function approaches as the input approaches a specific point. Understanding limits is essential for analyzing behavior near points where direct substitution may be undefined or indeterminate, such as h approaching 0 in this problem.

Exponential Limits and the Number e

Limits of the form (1 + kh)^{1/h} as h approaches 0 are closely related to the definition of the exponential constant e. Recognizing this pattern helps simplify expressions and evaluate limits involving powers that depend on a variable in the exponent.

L’Hôpital’s Rule

L’Hôpital’s Rule provides a method to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞ by differentiating the numerator and denominator separately. It is a powerful tool to confirm limit values obtained through numerical approximation or algebraic manipulation.