Limits Evaluate the following limits using Taylor series.
lim ₓ→∞ x(e¹/ˣ − 1)

L'Hôpital's Rule by Taylor series Suppose f and g have Taylor series about the point a.

a. If f(a) = g(a) = 0 and g′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent withl’Hôpital’s Rule.

b. If f(a) = g(a) =f′(a) = g′(a) = 0 and g′′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent with two applications of 1'Hôpital's Rule.

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

(1 + x⁴)⁻¹

Power series for derivatives

a. Differentiate the Taylor series centered at 0 for the following functions.

b. Identify the function represented by the differentiated series.

c. Give the interval of convergence of the power series for the derivative.

f(x) = (1 − x)⁻¹

Suppose f(0)=1, f'(0)=0, f''(0)=2, and f⁽³⁾(0)=6. Find the third-order Taylor polynomial for f centered at 0 and use it to approximate f(0.2).

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

√e

Limits Evaluate the following limits using Taylor series.

lim ₓ→₄ (x² 16)/(ln (x 3)}