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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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mathsLimit of a functionUnderstanding limits with Graphs

### Understanding Limits : Examples

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

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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
• <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
• >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 value of lim_(x->0) f(x)?

• =1
• =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 value of lim_(x->0) f(x)?

• oo
• >1
• <1
• =0
• =0

The answer is '=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 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
• =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.

Solved Exercise Problem:

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
• =0

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

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