Limits and Continuity Quiz

The formal definition of a limit of a function states that the function f has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L.

The concept of limit has many applications in modern calculus. In particular, the many definitions of continuity employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.

Bolzano introduced the basics of the epsilon-delta technique to define continuous functions in 1817.

<p>If some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist.</p> Signup and view all the answers

<p>In the calculus of one variable, the limiting value of the slope of secant lines to the graph of a function is the definition of the derivative.</p> Signup and view all the answers

<p>Ought to</p> Signup and view all the answers

<p>Infinitive form</p> Signup and view all the answers

<p>Hypothetical conditions</p> Signup and view all the answers

<p>-s</p> Signup and view all the answers

<p>'Do'</p> Signup and view all the answers