Math Doubts

Evaluate $\displaystyle \large \lim_{x \,\to\, 7}{\normalsize (4x)}$

The limit of the simple expression $4x$ has to calculate as $x$ approaches $7$ in this problem and this problem is useful for the beginners. The limit of the function $4x$ can be evaluated by substituting any value that is closer to $7$.

You can take any value that is closer to $7$, for example $x = 6.9917$. There are two reasons for this.

  1. The value of $6.9917$ is slightly less than $7$ but its approximate value is equal to $7$. In other words, $6.9917 \approx 7$.
  2. The difference between $7$ and $6.9917$ is also approximately small and negligible. In other words, $7-6.9917 = 0.0083$ and $0.0083 \approx 0$.

The above two points have cleared that the value of $6.9917$ closer to $7$ and substitute it to evaluate the limit of the function as $x$ approaches $7$.

$L \,=\, \displaystyle \large \lim_{x \,\to\, 7}{\normalsize (4x)}$

$\implies$ $L \,=\, 4(6.9917)$

$\implies$ $L \,=\, 4 \times 6.9917$

$\implies$ $L \,=\, 27.9668$

$\,\,\, \therefore \,\,\,\,\,\,$ $L \,\approx\, 28$

Therefore, the limit of the function $4x$ is equal to $28$ as $x$ approaches $7$.

It can also be obtained directly by substituting $x = 7$ in the given function.

$=\,\,\, 4(7)$

$=\,\,\, 4 \times 7$

$=\,\,\, 28$

Therefore, the limit of the function $4x$ as $x$ tends to $7$ is considered as the value of the function $4x$ at $x = 7$ in calculus.

$L \,=\, \displaystyle \large \lim_{x \,\to\, 7}{\normalsize (4x)}$

$\implies$ $L \,=\, 4(7)$

$\implies$ $L \,=\, 4 \times 7$

$\,\,\, \therefore \,\,\,\,\,\,$ $L \,=\, 28$

Theoretically, it is wrong to consider that the limit of the function $4x$ as $x$ approaches $7$ is equal to the value of the function at $x = 7$. However, it is acceptable to consider that they both are equal due to the negligible difference between the input values and also negligible difference between their corresponding values of the function.

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