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### Understanding limits with Graphs

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Limits by Numerator and Denominator

Slopes at the point x=a decide the limits of f(x) at x=a.

»  eg: f(x) = color(deepskyblue)(2x-4)/color(coral)(x-2)
→  slope of numerator at x=2 is 2

→  slope of denominator at x=2 is 1

→  Both LHL and RHL limits =2/1 = 2

### Understanding Limits : Examples

plain and simple summary

nub

plain and simple summary

nub

dummy

simple steps to build the foundation

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simple steps to build the foundation

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More examples are explained to evaluate limit of a function, using the graphs of the numerator and denominator.

Keep tapping on the content to continue learning.
Starting on learning "examples to understand limits" ;; More examples are explained to evaluate limit of a function, using the graphs of the numerator and denominator.

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x). What will be the value of lim_(x->2) f(x)?

• =1
• >1
• <1

The answer is '>1'. The slope of numerator is greater than the denominator.

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x). The function given is f(x) = color(deepskyblue)(2x-4)/color(coral)(x-2). The slope of the numerator is 2 and slope of the denominator is 1.

Left-hand-limit x=2-delta of
color(deepskyblue)(2x-4)/color(coral)(x-2)
quad quad = color(deepskyblue)(2(2-delta)-4)/color(coral)(2-delta-2)
quad quad = color(deepskyblue)(4-2delta-4)/color(coral)(-delta)
quad quad = color(deepskyblue)(-2delta)/color(coral)(-delta)
quad quad = 2

Similarly, the right-hand-limit can be worked out to 2.

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x). What will be the value of lim_(x->2) f(x)?

• =1
• <-1
• >1

The answer is '<-1'. The slope of the numerator is negative and decreasing steeper than the denominator.

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x). The function given is f(x) = color(deepskyblue)(6-3x)/color(coral)(.5x-1). The slope of the numerator is -3 and the slope of the denominator is 0.5.

Right-hand-limit x=2+delta
color(deepskyblue)(6-3(2+delta))/color(coral)(0.5(2+delta)-1)
quad quad = color(deepskyblue)(-3delta)/color(coral)(0.5delta)
quad quad = -6

Similarly, left hand limit can be worked out as -6.

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x). What will be the lim_(x->0) f(x)?

• =1
• >1
• <1

The answer is '=1'. The slope of numerator equals that of the denominator.

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x) (in purple). The function given is color(purple)(f(x)) = color(deepskyblue)(sin x)/color(coral)(x). At x=0 the slope of the numerator is 1 and slope of the denominator is 1.

Proof for limit of this function is explained later.

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x). What will be the lim_(x->0) f(x)?

• oo
• >1
• <1
• =0

The answer is '=0'. The slope of numerator is 0 and that of the denominator is 1.

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x) (in purple). The function given is f(x) = color(deepskyblue)(1- cos x)/color(coral)(x). The slope of the numerator is 0 and the slope of the denominator is 1.

Proof for limit of this function is explained later.

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x). What will be the lim_(x->0+) f(x)?

• =oo
• =1
• <1
• =0

The answer is '=oo'. This function is not a candidate to analyze numerator and denominator, as the function does not evaluate to 0/0 form.

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x) (in purple). The function given is f(x) = color(deepskyblue)(1)/color(coral)(x).
This function does not satisfy the pre-condition that numerator and denominator has to evaluate to 0. So the analysis by comparing slopes is not applicable.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Consider the graphs of numerator (in blue) and denominator (in orange) of function f(x). What will be the lim_(x->0) f(x)?

• oo
• >1
• <1
• =0

The answer is '0'. The slope of numerator is 0 and that of the denominator is 1.

Progress

Progress

Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 2, f of x?
equal;equals
equals 1
greater
greater than 1
less
less than 1
The answer is "greater than 1". The slope of numerator is greater than the denominator.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. ;; The function given is f of x = 2x minus 4 by x minus 2. The slope of the numerator is 2 and slope of the denominator is 1. ;; left hand limit x = 2 minus delta of 2 x minus 4 by x minus 2 ;; equals 2 into 2 minus delta minus 4, by , 2 minus delta minus 2;; equals 4 minus 2 delta minus 4, by, minus delta ;; equals minus 2 delta by minus delta;; equals 2;; Similarly, the right-hand-limit can be worked out to 2.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 2, f of x?
equals;equal
equals 1
less;minus
less than minus 1
greater
greater than 1
The answer is "less than minus 1". The slope of the numerator is negative and decreasing steeper than the denominator.
Consider the graphs of numerator in blue and denominator in orange of a function f of x.;; The function given is f of x = 6 minus 3 x divided by point 5 x minus 1. The slope of numerator is minus 3 and the slope of the denominator is point 5. Right hand limit is worked out as minus 6. Similarly, left hand limit can be worked out as minus 6.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 0, f of x?
equal;equals
equals 1
greater
greater than 1
less
less than 1
The answer is "equals 1". The slope of the numerator equals that of the denominator.
Consider the graphs of numerator in blue and denominator in orange of a function f of x in purple.;; The function given is f of x = sine x by x. ;; At x = 0; the slope of the numerator is 1 and the slope of the denominator is 1. ;; Proof for limit of this function is explained later.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 0, f of x?
infinity
infinity
greater
greater than 1
less
less than 1
equals;equal;0
equals 0
The answer is "equals 0". The slope of the numerator is 0 and that of the denominator is 1.
Consider the graphs of numerator in blue and denominator in orange of a function f of x in purple.;; the function given is f of x = 1 minus cos x divided by x. The slope of the numerator is 0 and the slope of the denominator is 1. ;; Proof for limit of this function is explained later.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 0+, f of x?
infinity
equals infinity
1
equals 1
less
less than 1
0
equals 0
The answer is "equals infinity. ". The function is not a candidate to analyze numerator and denominator, as the function does not evaluate to 0 by 0 form.
Consider the graphs of numerator in blue and denominator in orange of a function f of x in purple. ;; the function given is f of x = 1 by x. ;; This function does not satusfy the pre-condition that numerator and denominator has to evaluate to 0. So the analysis by comparing slopes is not applicable.
Consider the graphs of numerator in blue and denominator in orange of a function f of x. What will be the value of limit x tending to 0, f of x?
infinity
infinity
greater
greater than 1
less;1
less than 1
equals; equal;0
equals 0

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