Question

Find the following:

a) If
`\underset{x \rightarrow 5}{lim} (f(x) - 8)/(x - 5) = 4`, find
`\underset{x \rightarrow 5}{lim} f(x)`.
b) If
`\underset{x \rightarrow 5}{lim} (f(x) - 8)/(x - 5) = 7`, find
`\underset{x \rightarrow 5}{lim} f(x)`.

Answer 1

Explanation:

Limit refers to the value that the function approaches as the input approaches some value.

We say
`\displaystyle \lim_(x\rightarrow a)f(x)=L`, if f(x) approaches L as x approaches 'a'.

(a)

`\displaystyle \lim_(x\rightarrow 5)\left ( (f(x)-8)/(x-5) \right )=4\\(\displaystyle \lim_(x\rightarrow 5)f(x)-\displaystyle \lim_(x\rightarrow 5)8)/(\displaystyle \lim_(x\rightarrow 5)x-\displaystyle \lim_(x\rightarrow 5)5)=4\\`

`(\displaystyle \lim_(x\rightarrow 5)f(x)-8)/(\displaystyle \lim_(x\rightarrow 5)x-5)=4\\\displaystyle \lim_(x\rightarrow 5)f(x)-8=4\left ( \displaystyle \lim_(x\rightarrow 5)x-5 \right )\\\displaystyle \lim_(x\rightarrow 5)f(x)-8=4\displaystyle \lim_(x\rightarrow 5)x-4(5)\\\displaystyle \lim_(x\rightarrow 5)f(x)-8=4(5)-4(5)\\`

`\displaystyle \lim_(x\rightarrow 5)f(x)-8=20-20=0\\\displaystyle \lim_(x\rightarrow 5)f(x)=8`

(b)

`\displaystyle \lim_(x\rightarrow 5)\left ( (f(x)-8)/(x-5) \right )=7\\(\displaystyle \lim_(x\rightarrow 5)f(x)-\displaystyle \lim_(x\rightarrow 5)8)/(\displaystyle \lim_(x\rightarrow 5)x-\displaystyle \lim_(x\rightarrow 5)5)=7\\(\displaystyle \lim_(x\rightarrow 5)f(x)-8)/(\displaystyle \lim_(x\rightarrow 5)x-5)=7\\`

`\displaystyle \lim_(x\rightarrow 5)f(x)-8=7\left ( \displaystyle \lim_(x\rightarrow 5)x-5 \right )\\\displaystyle \lim_(x\rightarrow 5)f(x)-8=7\displaystyle \lim_(x\rightarrow 5)x-7(5)\\\displaystyle \lim_(x\rightarrow 5)f(x)-8=7(5)-7(5)\\\displaystyle \lim_(x\rightarrow 5)f(x)-8=35-35=0\\\displaystyle \lim_(x\rightarrow 5)f(x)=8`