## Functions of one variable, plane analytic geometry, and infinite series |

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Results 1-3 of 81

Page 87

□□m Example 2 Let g be defined by g(x) (a)

Find lim g(x) if it exists. ^TM Solution A sketch of the graph is shown in Fig. 2.3.2.

lim g(x) = lim (—x) = 0 and lim g(x) = lim x = 0 x->0~ x — 0~ x->0+ r->0 +

Therefore by Theorem 2.3.3, lim g(x) exists and is equal to 0. Note that x-0 g(0) =

2, which has no effect on lim g(x). Hx) = Example 3 Let h be defined by 4 - x2 ifx<l

2 + x2 if 1 < x (a)

limits if ...

□□m Example 2 Let g be defined by g(x) (a)

**Draw a sketch**of the graph of g. (b)Find lim g(x) if it exists. ^TM Solution A sketch of the graph is shown in Fig. 2.3.2.

lim g(x) = lim (—x) = 0 and lim g(x) = lim x = 0 x->0~ x — 0~ x->0+ r->0 +

Therefore by Theorem 2.3.3, lim g(x) exists and is equal to 0. Note that x-0 g(0) =

2, which has no effect on lim g(x). Hx) = Example 3 Let h be defined by 4 - x2 ifx<l

2 + x2 if 1 < x (a)

**Draw a sketch**of the graph of h. (b) Find each of the followinglimits if ...

Page 144

exist? (b) Is / continuous at 0? 50.

[x], and 0 < x < 2. (a) Does lim g(x) exist? (b) Is g continuous at 1? In Exercises 51

through 54, prove that the function is discontinuous at the number a. Then

determine if the discontinuity is removable or essential. If the discontinuity is

removable, define f(a) such that the discontinuity is removed. x2 + 2x - 8 r4 - x2 if

x < 1 1 53.

**Draw a sketch**of the graph of /if /(x) = [1 — x2] and — 2 < x < 2. (a) Does Um /(x)exist? (b) Is / continuous at 0? 50.

**Draw a sketch**of the graph of g if g(x) = (x — 1)[x], and 0 < x < 2. (a) Does lim g(x) exist? (b) Is g continuous at 1? In Exercises 51

through 54, prove that the function is discontinuous at the number a. Then

determine if the discontinuity is removable or essential. If the discontinuity is

removable, define f(a) such that the discontinuity is removed. x2 + 2x - 8 r4 - x2 if

x < 1 1 53.

Page 164

continuous on the closed interval [ — 2, 2]. Prove that neither /+( — 2) nor /'_(2)

exists.

continuous on ( — 00, —3] and [3, + 00). Prove that neither /'_( — 3) nor /'+(3)

exists.

continuous from the left at 2. Prove that /'_(2) does not exist.

graph of /. 21.

**Draw a sketch**of the graph of /. 18. Given /(a) = \/4 — a2, prove that / iscontinuous on the closed interval [ — 2, 2]. Prove that neither /+( — 2) nor /'_(2)

exists.

**Draw a sketch**of the graph of /. 19. Given/(A) = \/x2 — 9, prove that /iscontinuous on ( — 00, —3] and [3, + 00). Prove that neither /'_( — 3) nor /'+(3)

exists.

**Draw a sketch**of the graph of /. 20. Given /(a) = \/8 — x}, prove that / iscontinuous from the left at 2. Prove that /'_(2) does not exist.

**Draw a sketch**of thegraph of /. 21.

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### Contents

LIMITS AND CONTINUITY | 65 |

Review Exercises | 143 |

THE DIFFERENTIAL | 291 |

Copyright | |

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absolute maximum value absolute minimum absolutely convergent angle antiderivative axis chain rule closed interval concave downward concave upward continuous convergent coordinates cos2 cosh cosine cost function curve decreasing definite integral derivative differentiable directrix divergent domain Draw a sketch ellipse evaluate Example 2 Find exists f dx FIGURE Find an equation Find the area Find the volume focus following theorem formula ft/sec function defined graph is concave Hence hyperbolic hyperbolic cosine function ILLUSTRATION improper integral increasing inverse lim f(x Limit Theorem natural logarithmic obtain open interval parabola plane polar positive integer positive number power series Proof Prove radius rate of change real numbers rectangle region bounded relative maximum value Riemann sum sec2 sequence shown in Fig sin2 sinh slope solid of revolution Solution Let sq units subinterval Substituting tangent line velocity vertex vertices