Calculus 3A True/false

A tangent line intersects a function in exactly one point

FALSE: A tangent line may intersect the function a second time at a distant point. For example, x^3 ? x has almost every tangent line intersecting the graph at two points

A secant line intersects a function in at least two points

TRUE: A secant line is defined as the line connecting two points of a function. Thus, it must intersect the function in at least those two points.

If limx?a+ f(x) = limx?a? f(x) then limx?a f(x) exists

TRUE: If each side approaches the same number, then both sides together also approach that number

If limx?a+ f(x) and limx?a? f(x) both exist then limx?a f(x) exists

FALSE: It is possible for the two one-sided limits to exist but be different. For example, |x|/x has no limit at 0, since the two one-sided limits disagree

limx?0 1/x ? 1/(x^2 + x) = limx?0 1/x ? limx?0 1/(x^2 + x)

FALSE: The left side is a number limit, while the right side has two infinite limits which cannot be analyzed arithmetically. The limit laws only apply to finite limits.

limx?0 x/x + 1 = (limx?0 x)/(limx?0 x + 1)

TRUE: The limit laws say this is true as long as both limits exist and the denominator's limit is non-zero.

Every rational function is continuous everywhere

FALSE: A rational function can have infinite and removable discontinuities.

Every algebraic function is continuous everywhere it is defined

TRUE: Any discontinuity for an algebraic function is also a location where the function is undefined.

If f(x) ? g(x) for every x ?= a then limx?a f(x) ? limx?a g(x)

FALSE: The inequality is only true if both sides are defined, which is not always the case.

If f(x) ? g(x) for every x ? a then limx?a f(x) ? limx?a g(x) as long as both are defined

TRUE: This theorem is presented in the text.

If f(x) ? g(x) ? h(x) then limx?a g(x) exists

FALSE: Consider the functions ?1 ? sin(1/x) ? 1. The middle functions limit at 0 is undefined.

If f(x) ? g(x) ? h(x) and limx?a f(x) = limx?a h(x) then limx?a g(x) exists

TRUE: This is the statement of the squeeze theorem.

If f has a removable discontinuity then it differs from a continuous function at only one point

FALSE: There may be other discontinuities

If f has a removable discontinuity but is otherwise continuous then it differs from a continuous function at only one point

TRUE: The filling in the removable discontinuity would then make it continuous only changing a single value

If f has only infinite discontinuities, then it is continuous everywhere it is defined

FALSE: It may be defined in the infinite discontinuity. for example f(x) = 1/x if x ?= 0 and f(0) = 0

If f and g are continuous, then f + g is continuous

TRUE: This is one of the continuity laws

If f + g is continuous, then f is continuous

FALSE: Consider any function which is discontinuous, but defined everywhere. Then f + (?f ) = 0 is continuous, but f is not continuous.

Exponential functions are continuous

TRUE: Exponential functions are defined so as to be continuous everywhere

Piecewise functions are continuous if each piece is continuous

FALSE: The pieces must also agree on the boundaries of where they are defined

If f is continuous and invertible, then f^?1 is continuous

TRUE: This fact is what tells us that logarithms and inverse trig function are continuous

If f and g are continuous then f(g) is continuous

TRUE: This was presented in the text

If f(g) is continuous then f or g or both must be continuous

If f(g) is continuous then f or g or both must be continuous

If f(0)=0 and f(1)=1 then there exists a c such that
f(c) = 1/2

FALSE: Consider f(x) = 1 if x > 0 and f(x) = 0 otherwise. Then the conditions hold, but the function never takes on the value 1/2

If f is continuous then f(0) = 0 and f(1) = 1 then there exists a c such that f(c) = 1/2

TRUE: This follows from the squeeze theorem

A function has at most 2 horizontal asymptotes

TRUE: A horizontal asymptote can only occur at either positive or negative infinity. This means there can only be at most two

A rational function has only one horizontal asymptote

TRUE: The sign of the infinity does not impact the value of the asymptote of a rational function

sin(x) has two horizontal asymptotes

FALSE: sin(x) has no asymptotes, since it never settles towards a single value.

If f is continuous then f is differentiable

FALSE: Consider f(x) = |x|

If f is differentiable then f is continuous

TRUE: If f is discontinuous, then the limit defining the derivative cannot exist.