Consider the function and the limit .

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider the function

g(x)=\dfrac{\sqrt{x}-5}{x-25}

and the limit

L=\displaystyle\lim_{x\to 25} g(x)

. (a) Show analytically that direct substitution of

x=25

into

g(x)

leads to an indeterminate form. (b) Describe an algebraic manipulation that produces an equivalent expression for

g(x)

in which the indeterminacy is removed. Do not perform the entire computation; instead, explain the key algebraic step and why it resolves the difficulty identified in part (a). (c) After carrying out the manipulation you described in part (b), the limit

L

is found to exist and is finite. Explain, in complete sentences, how the removal of the indeterminate form guarantees the existence of

L

and what this implies about the local behavior of

\sqrt{x}

near

x=25

Related theory

Real Number System
Limits
Limits of algebraic functions
Limits and Derivatives
Limits and Continuity

Browse the problem library