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)`.

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)`. 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)`.

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)`. 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)`.

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). 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)`.

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). 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)`.

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). 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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