# A piecewise function is given. Use properties of limits to find the shown indicated limit, or state that the limit does not exist. f(x)=begin{cases}sqrt[3]{x^2+7}&text{if}&x<14x&text{if}&xgeq1end{cases} lim_{xrightarrow1} f(x)

Question
Limits and continuity
A piecewise function is given. Use properties of limits to find the shown indicated limit, or state that the limit does not exist. $$f(x)=\begin{cases}\sqrt[3]{x^2+7}&\text{if}&x<1\\4x&\text{if}&x\geq1\end{cases}$$</span>
$$\lim_{x\rightarrow1} f(x)$$

2021-02-04
Given
$$f(x)=\begin{cases}\sqrt[3]{x^2+7}&\text{if}&x<1\\4x&\text{if}&x\geq1\end{cases}$$</span>
we know for any given function g(x) limit at given point x=h exist only
if, $$\lim_{x->h^-}g(x)=\lim_{x->h^+}g(x),$$
thus, we are finding left hand limit and right hand limit one by one at x=1, if both are coming same then limit will exist other limit of given function will not exist
so,
$$L.H.L=\lim_{x\rightarrow1^-} f(x)$$
$$(\text{if }x<1,\text{then}f(x)=\sqrt[3]{x^2+7}$$</span>
$$=\lim_{x\rightarrow1^-}\sqrt[3]{x^2+7}$$
$$=\sqrt[3]{1^2+7}$$
$$=\sqrt[3]{1+7}$$
$$=\sqrt[3]{8}$$
$$=2$$
Now finding right hand limit also
$$R.H.L=\lim_{x\rightarrow1^+}f(x)$$
$$(\text{if } x>1\text{ then } f(x)=4x)$$
$$=\lim_{x\rightarrow1^+}(4x)$$
$$=\lim_{x\rightarrow^+}4(1)$$
$$=4$$
here
L.H.L != R.H.L.
hence, for given function limit at x=1 doesn't exists.

### Relevant Questions

Find the following limits or state that they do not exist. $$\lim_{x\rightarrow4}\frac{31x-42\cdot\sqrt{x+5}}{3-\sqrt{x+5}}$$
Find the following limits or state that they do not exist.
$$\lim_{h\rightarrow0}\frac{3}{\sqrt{16+3h}+4}$$
Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate.
$$\lim_{x\rightarrow\infty}\frac{4x^3-2x-1}{x^2-1}$$
Find the following limits or state that they do not exist. $$\lim_{w\rightarrow1}\frac{a\cdot1}{(w^2-2)}-\frac{1}{(w-1)b}$$
Evaluate the following limits or determine that they do not exist.
$$\lim_{(x,y,z)\rightarrow(2,2,3)}\frac{x^2z-3x^2-y^2z+3y^2}{xz-3x-yz+3y}$$
Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate.
$$\lim_{t\rightarrow0}\frac{5t^2}{\cos t-1}$$
Find the function The following limits represent the slope of a curve y=f(x) at the point {a,f(a)} Determine a possible function f and number a, then calculate the limit
$$\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}$$
Determine the limits if they exist: $$\lim_{(x,y)\rightarrow(2,4)}\frac{(x-2)^2(y-4)^2}{(x-2)^3+(y-4)^3}$$
Compute the following limits if they exist $$\lim_{x\rightarrow0}\frac{\sec x-1}{x^3}$$
$$\lim_{x\rightarrow1}\frac{1-x^2}{(x-1)^2}$$