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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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limit of linear slopes Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`.

The value of `lim_(x->2) f(x)>1`. This is because, the slope of numerator is greater than the denominator.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`.limit of linear slopes 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`.

limit of negative slopes Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`.

The value of `lim_(x->2) f(x) < -1`. This is because, 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)`.limit of negative slopes 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`.

limit of sin(x)/x slopes Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`.

The value of `lim_(x->0) f(x) = 1`. This is because, the slope of the numerator equals that of the denominator.

Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)` (in purple).limit of sin(x)/x slopes 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.

limit of (1-cos(x))/x slopes Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`.

The value of `lim_(x->0) f(x)=0`. This is because, 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).limit of (1-cos(x))/x slopes 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.

limit of 1/x slopes Consider the graphs of numerator (in blue) and denominator (in orange) of function `f(x)`.

The value of `lim_(x->0+) f(x)=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).limit of linear slopes 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)`. limit of (1-cos(x))/x slopes 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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