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### Understanding limits with Graphs

Voice

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Home

Limit of Functions with graph of numerator and denominator

»  Slopes at the point x=a decide the limits of f(x) at x=a.
→  slope of numerator at x=0 is 1

→  slope of denominator at x=0 is 1

→  Both LHL and RHL limits =1/1 = 1

### Understanding limits with Graphs of Numerator and Denominator

plain and simple summary

nub

plain and simple summary

nub

dummy

Slope of numerator and denominator defines the limits of the function.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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The function for which limit is computed is considered as two constituent functions of numerator and denominator. The limit of the function is explained with the graphs of numerator and denominator.

Keep tapping on the content to continue learning.
Starting on learning "Understanding limits with Graphs of Numerator and Denominator". ;; The function for which limit is computed is considered as two constituent functions of numerator and denominator. The limit of the function is explained with the graphs of numerator and denominator.

The limit of a function is computed when the function evaluates to indeterminate value 0/0 at x=a. Seeing the division in indeterminate value 0/0, the function can be given as f(x)=color(deepskyblue)(f_n(x))/color(coral)(f_d(x)) such that color(deepskyblue)(f_n(x))|_(x=a) = 0 and color(coral)(f_d(x))|_(x=a) = 0.

Consider the function f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1). The numerator and denominator evaluate to 0 at x=1. Could you identify which graph represents the numerator and which graph represents the denominator?

• numerator is the curve in blue, and denominator is the line in orange
• numerator is the line in orange, and denominator is the curve in blue

The answer is ' numerator is the curve in blue, and denominator is the line in orange'

Given the graphs of numerator and denominator of the function f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1). The vertical purple lines, show 1-delta and 1+delta. What will be the value of denominator at 1-delta and 1+delta?

• color(coral)(-delta) and color(coral)(+delta)
• color(coral)(+delta) and color(coral)(-delta)

The answer is '-delta and +delta'.

Given the graphs of numerator and denominator of the function f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1). The vertical purple lines, show 1-delta and 1+delta. What will be the value of numerator at 1-delta and 1+delta?

• color(deepskyblue)(+2delta+delta^2) and color(deepskyblue)(-2delta+delta^2)
• color(deepskyblue)(-2delta+delta^2) and color(deepskyblue)(+2delta+delta^2)

The answer is '-2delta+delta^2 and +2delta+delta^2'.

Given the graphs of numerator and denominator of the function f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1). The vertical purple lines, show 1-delta and 1+delta. What parameter of the line defines the values of denominator at 1-delta and 1+delta?

• slope of the line
• position of the line

The answer is 'slope of the line'.

Given the graphs of numerator and denominator of the function f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1). The vertical purple lines, show 1-delta and 1+delta. Considering the curve at position x=1 as piecewise linear, What parameter of the curve defines the values of numerator at 1-delta and 1+delta?

• slope of the curve at that position x=1
• position of the line in the graph

The answer is 'slope of the curve at that position'.

Given the function f(x)=color(deepskyblue)(f_n(x))/color(coral)(f_d(x)) such that color(deepskyblue)(f_n(x))|_(x=a) = 0 and color(coral)(f_d(x))|_(x=a) = 0.

f(x)|_(x=a+delta)
quad quad = color(deepskyblue)(f_n(x))|_(x=a+delta) -: color(coral)(f_d(x))|_(x=a+delta)
quad quad = [color(deepskyblue)(f_n(x))|_(x=a+delta) - f_n(a)]
quad quad quad quad -: [color(coral)(f_d(x))|_(x=a+delta) - f_d(a)]
as f_n(a) = 0 and f_d(a)=0.

quad quad = [color(deepskyblue)(f_n(x))|_(x=a+delta) - f_n(a)]/delta
quad quad quad quad -: [color(coral)(f_d(x))|_(x=a+delta) - f_d(a)]/delta
quad quad = color(deepskyblue)(text(slope) f_n(x)|_(x=a)) -: color(coral)(text(slope) f_d(x)|_(x=a)).

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Geometrical representation of Limits: If f(x)=color(deepskyblue)(f_n(x))/color(coral)(f_d(x)), where
f(x)|_(x=a) = 0/0;
color(deepskyblue)(f_n(x))|_(x=a) = 0 and
color(coral)(f_d(x))|_(x=a) = 0, then

•  the slopes on the left of x=a define the left-hand-limit and

•  the slopes on the right of x=a define the right-hand-limit.

The slopes referred are for the numerator and denominator.

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

The limit of a function is computed when the function evaluates to indeterminate value 0 by 0 at x=a. Seeing the division in indeterminate value 0 by 0 , the function can be given as f of x = f n of x divided by f d of x, such that f n of x at x = a equals 0, and f d of x at x = a equals 0. ;; f n of x is the numerator;; f d of x is the denominator;;
Consider the function f of x = x squared minus 1 by x minus 1. The numerator and denominator evaluate to 0 at x = 1. Could you identify which graph represents the numerator and which graph represents the denominator?
1
2
The answer is ' numerator is the curve in blue, and denominator is the line in orange'
Given the graphs of numerator and denominator of the function f of x = x squared minus 1 by x minus 1. The vertical purple lines, show 1 minus delta and 1+delta. What will be the value of denominator at 1 minus delta and 1+delta?
1
minus delta and plus delta
2
plus delta and minus delta
The answer is "minus delta and plus delta"
Given the graphs of numerator and denominator of the function f of x = x squared minus 1 by x minus 1. The vertical purple lines, show 1 minus delta and 1+delta. What will be the value of numerator at 1 minus delta and 1+delta?
1
plus 2 delta + delta squared and minus 2 delta + delta squared
2
minus 2 delta + delta squared and plus 2 delta + delta squared
The answer is "minus 2 delta + delta squared and plus 2 delta + delta squared"
Given the graphs of numerator and denominator of the function f of x = x squared minus 1 by x minus 1. The vertical purple lines, show 1 minus delta and 1+delta. ;; What parameter of the line defines the values of denominator at 1 minus delta and 1+delta?
slope
slope of the line
position
position of the line