Question

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 + $$2h)^{1/h}$$

Accepted Answer

Recognize that the limit is of the form $$\lim_{h 	o 0} (1 + 2h$$)^{\frac{1}$${h}}$$, which resembles an expression that can be related to the exponential function $$e^x$$ through limits of the form $$(1 + k h$$)^{\frac{1}$${h}} as$$ $$h 	o 0. To $$approximate the limit using a calculator, create a table of values for $$h$$ approaching 0 from both the 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 $$(1 + 2h$$)^{\frac{1}$${h}}. $$Observe the trend of the values in the table as $$h$$ gets closer to 0 to estimate the limit numerically. To confirm the result analytically using l’Hôpital’s Rule, first rewrite the limit in a form suitable for applying the rule by taking the natural logarithm: consider $$L = \lim_{h 	o 0} (1 + 2h$$)^{\frac{1}$${h}}$$, then $$\ln L = \lim_{h 	o 0} \frac{\ln(1 + 2h)}{h}. $$Apply l’Hôpital’s Rule to the limit $$\lim_{h 	o 0} \frac{\ln(1 + 2h)}{h} by $$differentiating numerator and denominator with respect to $$h$$, then evaluate the resulting limit to find $$\ln L. $$Finally, exponentiate to find $$L.$$

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 + 2h)^{1/h}

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Key Concepts

Limit of a Function

Exponential and Logarithmic Limits

L’Hôpital’s Rule

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 + 2h)^{1/h}