Calculus (Math 1A) Lecture 10 Vivek Shende September 15, 2017 - PowerPoint PPT Presentation

Calculus (Math 1A) Lecture 10 Vivek Shende September 15, 2017

Hello and welcome to class!

Hello and welcome to class! Last time

Hello and welcome to class! Last time We discussed continuity.

Hello and welcome to class! Last time We discussed continuity. Today

Hello and welcome to class! Last time We discussed continuity. Today We will define limits at infinity, and give examples of their computation.

Asymptotes of a graph

Asymptotes of a graph

Asymptotes of a graph Asymptotes are lines which the graph approaches.

Vertical asymptotes

Vertical asymptotes A vertical asymptote is a line x = a where x → a + f ( x ) = ±∞ lim or x → a − f ( x ) = ±∞ lim

Horizontal asymptotes

Horizontal asymptotes A horizontal asymptote is a line y = b where x →∞ f ( x ) = b lim or x →−∞ f ( x ) = b lim

Horizontal asymptotes A horizontal asymptote is a line y = b where x →∞ f ( x ) = b lim or x →−∞ f ( x ) = b lim We’ll say what these limits mean shortly.

More asymptotes

More asymptotes

More asymptotes

More asymptotes We won’t discuss these in this class.

Why asymptotes?

Why asymptotes? Asymptotes know qualitative and large-scale behavior.

Why asymptotes? Asymptotes know qualitative and large-scale behavior. Vertical asymptotes tell you where the function gets large.

Why asymptotes? Asymptotes know qualitative and large-scale behavior. Vertical asymptotes tell you where the function gets large. Horizontal asymptotes tell you what happens when x gets large.

Why asymptotes? Asymptotes know qualitative and large-scale behavior. Vertical asymptotes tell you where the function gets large. Horizontal asymptotes tell you what happens when x gets large. You can also learn about roots, using intermediate value theorem:

Limits at infinity

Limits at infinity x →∞ f ( x ) = y lim

Limits at infinity x →∞ f ( x ) = y lim Informally: f ( x ) approaches y as x approaches infinity.

Limits at infinity x →∞ f ( x ) = y lim Informally: f ( x ) approaches y as x approaches infinity. More precisely: f ( x ) can be brought within any fixed distance of y by ensuring that x is sufficiently large.

Limits at infinity x →∞ f ( x ) = y lim Informally: f ( x ) approaches y as x approaches infinity. More precisely: f ( x ) can be brought within any fixed distance of y by ensuring that x is sufficiently large. In symbols:

Limits at infinity x →∞ f ( x ) = y lim Informally: f ( x ) approaches y as x approaches infinity. More precisely: f ( x ) can be brought within any fixed distance of y by ensuring that x is sufficiently large. In symbols: for every ǫ > 0,

Limits at infinity x →∞ f ( x ) = y lim Informally: f ( x ) approaches y as x approaches infinity. More precisely: f ( x ) can be brought within any fixed distance of y by ensuring that x is sufficiently large. In symbols: for every ǫ > 0, there exists some N > 0 such that if x > N then | f ( x ) − y | < ǫ .

Infinite limits at infinity

Infinite limits at infinity x →∞ f ( x ) = ∞ lim

Infinite limits at infinity x →∞ f ( x ) = ∞ lim Informally: f ( x ) approaches infinity as x approaches infinity.

Infinite limits at infinity x →∞ f ( x ) = ∞ lim Informally: f ( x ) approaches infinity as x approaches infinity. More precisely: f ( x ) can be made arbitrarily large by ensuring that x is sufficiently large.

Infinite limits at infinity x →∞ f ( x ) = ∞ lim Informally: f ( x ) approaches infinity as x approaches infinity. More precisely: f ( x ) can be made arbitrarily large by ensuring that x is sufficiently large. In symbols:

Infinite limits at infinity x →∞ f ( x ) = ∞ lim Informally: f ( x ) approaches infinity as x approaches infinity. More precisely: f ( x ) can be made arbitrarily large by ensuring that x is sufficiently large. In symbols: for every M > 0,

Infinite limits at infinity x →∞ f ( x ) = ∞ lim Informally: f ( x ) approaches infinity as x approaches infinity. More precisely: f ( x ) can be made arbitrarily large by ensuring that x is sufficiently large. In symbols: for every M > 0, there exists some N > 0 such that if x > N then f ( x ) > M .

Limits at infinity from one-sided limits

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim y → 0 + f (1 / y )

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim y → 0 + f (1 / y ) Asserting the left hand side is equal to some value c is saying:

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim y → 0 + f (1 / y ) Asserting the left hand side is equal to some value c is saying:

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim y → 0 + f (1 / y ) Asserting the left hand side is equal to some value c is saying: for every ǫ , there’s a N such that x > N implies | f ( x ) − c | < ǫ .

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim y → 0 + f (1 / y ) Asserting the left hand side is equal to some value c is saying: for every ǫ , there’s a N such that x > N implies | f ( x ) − c | < ǫ . Asserting the left hand side is equal to some value c is saying:

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim y → 0 + f (1 / y ) Asserting the left hand side is equal to some value c is saying: for every ǫ , there’s a N such that x > N implies | f ( x ) − c | < ǫ . Asserting the left hand side is equal to some value c is saying:

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim y → 0 + f (1 / y ) Asserting the left hand side is equal to some value c is saying: for every ǫ , there’s a N such that x > N implies | f ( x ) − c | < ǫ . Asserting the left hand side is equal to some value c is saying: for every ǫ , there’s a δ such that 0 < y < δ implies | f (1 / y ) − c | < ǫ .

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim y → 0 + f (1 / y ) Asserting the left hand side is equal to some value c is saying: for every ǫ , there’s a N such that x > N implies | f ( x ) − c | < ǫ . Asserting the left hand side is equal to some value c is saying: for every ǫ , there’s a δ such that 0 < y < δ implies | f (1 / y ) − c | < ǫ . To go between one and the other, take y = 1 / x and δ = 1 / N .

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim x → 0 + f (1 / x )

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim x → 0 + f (1 / x ) It follows that the limit laws we learned before hold for limits at infinity, except the ones involving plugging in the value.

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim x → 0 + f (1 / x ) It follows that the limit laws we learned before hold for limits at infinity, except the ones involving plugging in the value. Also, for n > 0:

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim x → 0 + f (1 / x ) It follows that the limit laws we learned before hold for limits at infinity, except the ones involving plugging in the value. x →∞ x − n Also, for n > 0: we have lim

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim x → 0 + f (1 / x ) It follows that the limit laws we learned before hold for limits at infinity, except the ones involving plugging in the value. x →∞ x − n = lim Also, for n > 0: we have lim x → 0 + x n

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim x → 0 + f (1 / x ) It follows that the limit laws we learned before hold for limits at infinity, except the ones involving plugging in the value. x →∞ x − n = lim x → 0 + x n = 0 Also, for n > 0: we have lim

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim x → 0 + f (1 / x ) It follows that the limit laws we learned before hold for limits at infinity, except the ones involving plugging in the value. x →∞ x − n = lim x → 0 + x n = 0 Also, for n > 0: we have lim and lim x →∞ x n

Limits at infinity from one-sided limits By comparing definitions, one has: x →∞ f ( x ) = lim lim x → 0 + f (1 / x ) It follows that the limit laws we learned before hold for limits at infinity, except the ones involving plugging in the value. x →∞ x − n = lim x → 0 + x n = 0 Also, for n > 0: we have lim x →∞ x n = lim and lim x → 0 + x − n