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When the limit of function Doesnot exist and when is it infinity or -infinity

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Here we need to use 3+. which means x is little more than 3. Example 3.000001 or 3.001 or 3.02 or a number close to 3 but little larger than 3 Therefore, if we plug in x= 3.000001 in the function, we get  and if we keep reducing the value of x but ensuring it is still larger than 3, we can observe that the final answer keeps increasing and getting really large or  Examples of such values of x are:3.000000000000000000005 or 3.0000000000000000000001 You can plug in the above values and verify the result to be really large. and hence the Case2: Here it is 3-, which means x is close to 3 but little less than 3 Try to think of such numbers! Anyways, Examples of x are 2.99999 or 2.99998 and infinitely more. Lets plug in one of such values of x in the expression which is close to 3 but less than 3, Usin x=2.99999 in the expression and if we keep increasing the value of x but ensuring it is still less than 3, we can observe that the final answer keeps decreasing and getting really sma...
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Example2: Graphing |f(|x|)|

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 f(x) =-x^3 +x^2 Step 1: Graphing. |f(x)| from f(x) So, reflecting the part of f(x) below X-Axis to above it Step2: f(|x|) from |f(x)| Getting rid of |f(x)| on the left side of the X-axis  Finally f(|x|) from |f(x)|  By Reflecting the right side of |f(x)| on the left side of X-axis i.e |f(|x|) graph
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Graphing |f(|x|)| from f(x)

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Graphing |f(|x|)| from f(x) can be broken into two steps f(x)=sinx Step1 : f(|x|) from f(x) Getting rid of the f(x) on negative X-Axis and f(x) = f(|x|) on the positive or right side of X-Axis. Reflecting f(|x|) on the left side f(|x|) Graph is below Step2: Sketching |f(x)| using f(|x|) i.e. reflecting the part of f(|x|) that is below the X-Axis , above the X-Axis PS: |f(x)| = |f(|x|)) in this case, it may not true in all cases
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Graphing |f(x)| using f(x)

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 f(x) is the output or the y-value of the function. |f(x)| therefore implies the modulus of the function itself Example if f(x) =4. |f(x)| = 4 and if  f(x) = -4, then |f(x)| =4 Hence , |f(x)| must lie above the X-axis. All the negative part of f(x) is reflected above the X-Axis to get |f(x)| f(x) = sinx |f(x)| = |sinx|
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f(x) = |x|

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f(x) = |x| is a machine which accepts all kinds of numbers - positive, negative and 0 The positive numbers or 0, when enter the machine come out of it, without a change But when the negative numbers enter the machine, they are multiplied with -1 so as to make them positive. so, guess what happens when the input is "x"? What sign is x? What happens if x is positive? and What happens if x is negative?
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Graphing f|x| from f(x)

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The input of f(x) is x Whereas the input of f(|x|) is |x|  x=|x| when x is positive. Example f(x) = sinx f(|x|) = sin(|x|) Therefore graph of f(x) is same as that of f(|x|) for positive values of x only i.e on the right side of X-Axis x =-|x| for negative values of x. That implies if we input negative numbers in f(|x|), we end up giving -x as input and since x is negative in this case, -x is a positive number with the same value. Example when x=2 f(2) = f(|2|)  But if x=-2 f(|-2|) =f(2) Hence f(2)= f(|2|) = f(|-2|) When x=-3 f(|-3|) = f(3)  For x=-4 f(|-4|)=f(4) Whatever number, we may use, it turns positive before entering f(|x|) and hence is determined by f(x) where x is positive number Therefore f(|x|) is determined by f(x) with only positive inputs. For f(|x|) f(x)= f(-x) Hence whatever shape is f(|x|) is on right side of X-Axis as sketched previously, is on left side as well. f(|x|) is an even function.
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Within or without

 While the saints and all our ancient holy scriptures urge us to look within. Be it to find God or to cleanse your mind. All the faults lie within If the table shakes on which the cup of tea lies, it spills tea because it contains tea not because someone shook the table. That is how the perspectives are shaped when we follow saints. This is an honest truth that not everyone follows saints in the modern day and when democracy prevails. Some would like to think tea spilled because someone shook the table which is also correct in another frame. Whatever lies within the cup is none of our business, just don't spill it! What if democracy has reached a point when the "without" perspective of spilling the tea is widely prevalent? The essence of democracy is lost and it becomes dadagiri because we start demanding freedom that compromises the other's freedom. While democracy promotes the without perspective, the within perspective must be preserved to keep the democracy going....
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